MODERN CANONICAL QUANTUMGENERAL RELATIVITY

Modern physics rests on two fundamental building blocks: general relativity andquantum theory. General relativity is a geometric interpretation of gravity, whilequantum theory governs the microscopic behaviour of matter. According to Ein-steins equations, geometry is curved when and where matter is localized. There-fore, in general relativity, geometry is a dynamical quantity that cannot be pre-scribed a priori but is in interaction with matter. The equations of nature arebackground independent in this sense; there is no space-time geometry on whichmatter propagates without backreaction of matter on geometry. Since matter isdescribed by quantum theory, which in turn couples to geometry, we need a quan-tum theory of gravity. The absence of a viable quantum gravity theory to date isdue to the fact that quantum (eld) theory as currently formulated assumes thata background geometry is available, thus being inconsistent with the principles ofgeneral relativity. In order to construct quantum gravity, one must reformulatequantum theory in a background-independent way. Modern Canonical QuantumGeneral Relativity is about one such candidate for a background-independentquantum gravity theory: loop quantum gravity.

This book provides a complete treatise of the canonical quantization of gen-eral relativity. The focus is on detailing the conceptual and mathematical frame-work, describing the physical applications, and summarizing the status of thisprogramme in its most popular incarnation: loop quantum gravity. Mathemat-ical concepts and their relevance to physics are provided within this book, soit is suitable for graduate students and researchers with a basic knowledge ofquantum eld theory and general relativity.

T homas Th i emann is Sta Scientist at the Max Planck Institut furGravitationsphysik (Albert Einstein Institut), Potsdam, Germany. He is alsoa long-term researcher at the Perimeter Institute for Theoretical Physics andAssociate Professor at the University of Waterloo, Canada. Thomas Thiemannobtained his Ph.D. in theoretical physics from the Rheinisch-Westfalisch Tech-nische Hochschule, Aachen, Germany. He held two-year postdoctoral positions atThe Pennsylvania State University and Harvard University. As of 2005 he holdsa guest professor position at Beijing Normal University, China.

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Copyright: Max Planck Institute for Gravitational Physics (Albert Einstein Insti-tute), MildeMarketing Science Communication, Exozet. To see the animation, please visit theURL http://www.einstein-online.info/de/vertiefung/Spinnetzwerke/ index.html.

Quantum spin dynamicsThis is a still from an animation which illustrates the dynamical evolution of quantum geometryin Loop Quantum Gravity (LQG), which is a particular incarnation of canonical QuantumGeneral Relativity.

The faces of the tetrahedra are elementary excitations (atoms) of geometry. Each face iscoloured, where red and violet respectively means that the face carries low or high area respec-tively. The colours or areas are quantised in units of the Planck area 2P 10 66 cm2 . Thusthe faces do not have area as they appear to have in the gure, rather one would have to shrinkred 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 whichconnects the centres of the tetrahedra with a common face. These edges are charged withhalf-integral spin-quantum numbers and these numbers are proportional to the quantum areaof the faces. The collection of spins and edges denes a spin-network state. The spin quantumnumbers are created and annihilated at each Planck time step of P 10 43 s in a specicway 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 anni-hilated all the time. Therefore, the free space not occupied by tetrahedra does not correspondto empty (matter-free) space but rather to space without geometry, it has zero volume andtherefore is a hole in the quantum spacetime. The tetrahedra are not embedded in space, theyare 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 thePlanck scale, only on large scales does it appear smooth.

In this book, this fascinating physics is explained in mathematical detail.

Foreword, by Chris Isham pagexviiPreface xixNotation and conventions xxiii

Introduction: Dening quantum gravity 1Why quantum gravity in the twenty-rst century? 1The role of background independence 8Approaches to quantum gravity 11Motivation for canonical quantum general relativity 23Outline of the book 25

I CLASSICAL FOUNDATIONS, INTERPRETATION AND THECANONICAL QUANTISATION PROGRAMME

1 Classical Hamiltonian formulation of General Relativity 391.1 The ADM action 391.2 Legendre transform and Dirac analysis of constraints 461.3 Geometrical interpretation of the gauge transformations 501.4 Relation between the four-dimensional dieomorphism group and

the transformations generated by the constraints 561.5 Boundary conditions, gauge transformations and symmetries 60

1.5.1 Boundary conditions 601.5.2 Symmetries and gauge transformations 65

2 The problem of time, locality and the interpretation ofquantum mechanics 74

2.1 The classical problem of time: Dirac observables 752.2 Partial and complete observables for general constrained systems 81

2.2.1 Partial and weak complete observables 822.2.2 Poisson algebra of Dirac observables 852.2.3 Evolving constants 892.2.4 Reduced phase space quantisation of the algebra of Dirac

observables and unitary implementation of themulti-ngered time evolution 90

2.3 Recovery of locality in General Relativity 93

x Contents

2.4 Quantum problem of time: physical inner product andinterpretation of quantum mechanics 952.4.1 Physical inner product 952.4.2 Interpretation of quantum mechanics 98

3 The programme of canonical quantisation 1073.1 The programme 108

4 The new canonical variables of Ashtekar forGeneral Relativity 118

4.1 Historical overview 1184.2 Derivation of Ashtekars variables 123

4.2.1 Extension of the ADM phase space 1234.2.2 Canonical transformation on the extended phase space 126

II FOUNDATIONS OF MODERN CANONICAL QUANTUMGENERAL RELATIVITY

5 Introduction 1415.1 Outline and historical overview 141

6 Step I: the holonomyux algebra P 1576.1 Motivation for the choice of P 1576.2 Denition of P: (1) Paths, connections, holonomies and

cylindrical functions 1626.2.1 Semianalytic paths and holonomies 1626.2.2 A natural topology on the space of generalised connections 1686.2.3 Gauge invariance: distributional gauge transformations 1756.2.4 The C algebraic viewpoint and cylindrical functions 183

6.3 Denition of P: (2) surfaces, electric elds, uxes and vector elds 1916.4 Denition of P: (3) regularisation of the holonomyux

Poisson algebra 1946.5 Denition of P: (4) Lie algebra of cylindrical functions and

ux vector elds 202

7 Step II: quantum -algebra A 2067.1 Denition of A 2067.2 (Generalised) bundle automorphisms of A 209

8 Step III: representation theory of A 2128.1 General considerations 2128.2 Uniqueness proof: (1) existence 219

8.2.1 Regular Borel measures on the projective limit:the uniform measure 220

8.2.2 Functional calculus on a projective limit 226

Contents xi

8.2.3 + Density and support properties of A,A/G with respectto A,A/G 233

8.2.4 Spin-network functions and loop representation 2378.2.5 Gauge and dieomorphism invariance of 0 2428.2.6 + Ergodicity of 0 with respect to spatial dieomorphisms 2458.2.7 Essential self-adjointness of electric ux momentum

operators 2468.3 Uniqueness proof: (2) uniqueness 2478.4 Uniqueness proof: (3) irreducibility 252

9 Step IV: (1) implementation and solution of thekinematical constraints 264

9.1 Implementation of the Gau constraint 2649.1.1 Derivation of the Gau constraint operator 2649.1.2 Complete solution of the Gau constraint 266

9.2 Implementation of the spatial dieomorphism constraint 2699.2.1 Derivation of the spatial dieomorphism constraint

operator 2699.2.2 General solution of the spatial dieomorphism constraint 271

10 Step IV: (2) implementation and solution of theHamiltonian constraint 279

10.1 Outline of the construction 27910.2 Heuristic explanation for UV niteness due to background

independence 28210.3 Derivation of the Hamiltonian constraint operator 28610.4 Mathematical denition of the Hamiltonian constraint operator 291

10.4.1 Concrete implementation 29110.4.2 Operator limits 29610.4.3 Commutator algebra 30010.4.4 The quantum Dirac algebra 309

10.5 The kernel of the WheelerDeWitt constraint operator 31110.6 The Master Constraint Programme 317

10.6.1 Motivation for the Master Constraint Programme inGeneral Relativity 317

10.6.2 Denition of the Master Constraint 32010.6.3 Physical inner product and Dirac observables 32610.6.4 Extended Master Constraint 32910.6.5 Algebraic Quantum Gravity (AQG) 331

10.7 + Further related results 33410.7.1 The Wick transform 33410.7.2 Testing the new regularisation technique by models of

quantum gravity 340

xii Contents

10.7.3 Quantum Poincare algebra 34110.7.4 Vasiliev invariants and discrete quantum gravity 344

11 Step V: semiclassical analysis 34511.1 + Weaves 34911.2 Coherent states 353

11.2.1 Semiclassical states and coherent states 35411.2.2 Construction principle: the complexier method 35611.2.3 Complexier coherent states for dieomorphism-invariant

theories of connections 36211.2.4 Concrete example of complexier 36711.2.5 Semiclassical limit of loop quantum gravity: graph-changing

operators, shadows and dieomorphism-invariantcoherent states 376

11.2.6 + The innite tensor product extension 38511.3 Graviton and photon Fock states from L2(A, d0) 390

III PHYSICAL APPLICATIONS

12 Extension to standard matter 39912.1 The classical standard model coupled to gravity 400

12.1.1 Fermionic and Einstein contribution 40112.1.2 YangMills and Higgs contribution 405

12.2 Kinematical Hilbert spaces for dieomorphism-invariant theoriesof fermion and Higgs elds 40612.2.1 Fermionic sector 40612.2.2 Higgs sector 41112.2.3 Gauge and dieomorphism-invariant subspace 417

12.3 Quantisation of matter Hamiltonian constraints 41812.3.1 Quantisation of EinsteinYangMills theory 41912.3.2 Fermionic sector 42212.3.3 Higgs sector 42512.3.4 A general quantisation scheme 429

13 Kinematical geometrical operators 43113.1 Derivation of the area operator 43213.2 Properties of the area operator 43413.3 Derivation of the volume operator 43813.4 Properties of the volume operator 447

13.4.1 Cylindrical consistency 44713.4.2 Symmetry, positivity and self-adjointness 44813.4.3 Discreteness and anomaly-freeness 44813.4.4 Matrix elements 449

13.5 Uniqueness of the volume operator, consistency with the uxoperator and pseudo-two-forms 453

Contents xiii

13.6 Spatially dieomorphism-invariant volume operator 455

14 Spin foam models 45814.1 Heuristic motivation from the canonical framework 45814.2 Spin foam models from BF theory 46214.3 The BarrettCrane model 466

14.3.1 Plebanski action and simplicity constraints 46614.3.2 Discretisation theory 47214.3.3 Discretisation and quantisation of BF theory 47614.3.4 Imposing the simplicity constraints 48214.3.5 Summary of the status of the BarrettCrane model 494

14.4 Triangulation dependence and group eld theory 49514.5 Discussion 502

15 Quantum black hole physics 51115.1 Classical preparations 514

15.1.1 Null geodesic congruences 51415.1.2 Event horizons, trapped surfaces and apparent horizons 51715.1.3 Trapping, dynamical, non-expanding and (weakly) isolated

horizons 51915.1.4 Spherically symmetric isolated horizons 52615.1.5 Boundary symplectic structure for SSIHs 535

15.2 Quantisation of the surface degrees of freedom 54015.2.1 Quantum U(1) ChernSimons theory with punctures 541

15.3 Implementing the quantum boundary condition 54615.4 Implementation of the quantum constraints 548

15.4.1 Remaining U(1) gauge transformations 54915.4.2 Remaining surface dieomorphism transformations 55015.4.3 Final physical Hilbert space 550

15.5 Entropy counting 55015.6 Discussion 557

16 Applications to particle physics and quantum cosmology 56216.1 Quantum gauge xing 56216.2 Loop Quantum Cosmology 563

17 Loop Quantum Gravity phenomenology 572

IV MATHEMATICAL TOOLS AND THEIR CONNECTIONTO PHYSICS

18 Tools from general topology 57718.1 Generalities 57718.2 Specic results 581

xiv Contents

19 Dierential, Riemannian, symplectic and complexgeometry 585

19.1 Dierential geometry 58519.1.1 Manifolds 58519.1.2 Passive and active dieomorphisms 58719.1.3 Dierential calculus 590

19.2 Riemannian geometry 60619.3 Symplectic manifolds 614

19.3.1 Symplectic geometry 61419.3.2 Symplectic reduction 61619.3.3 Symplectic group actions 621

19.4 Complex, Hermitian and Kahler manifolds 623

20 Semianalytic category 62720.1 Semianalytic structures on Rn 62720.2 Semianalytic manifolds and submanifolds 631

21 Elements of bre bundle theory 63421.1 General bre bundles and principal bre bundles 63421.2 Connections on principal bre bundles 636

22 Holonomies on non-trivial bre bundles 64422.1 The groupoid of equivariant maps 64422.2 Holonomies and transition functions 647

23 Geometric quantisation 65223.1 Prequantisation 65223.2 Polarisation 66223.3 Quantisation 668

24 The Dirac algorithm for eld theories with constraints 67124.1 The Dirac algorithm 67124.2 First- and second-class constraints and the Dirac bracket 674

25 Tools from measure theory 68025.1 Generalities and the RieszMarkov theorem 68025.2 Measure theory and ergodicity 687

26 Key results from functional analysis 68926.1 Metric spaces and normed spaces 68926.2 Hilbert spaces 69126.3 Banach spaces 69326.4 Topological spaces 69426.5 Locally convex spaces 69426.6 Bounded operators 69526.7 Unbounded operators 697

Contents xv

26.8 Quadratic forms 699

27 Elementary introduction to Gelfand theory forAbelian C-algebras 701

27.1 Banach algebras and their spectra 70127.2 The Gelfand transform and the Gelfand isomorphism 709

28 Bohr compactication of the real line 71328.1 Denition and properties 71328.2 Analogy with loop quantum gravity 715

29 Operator -algebras and spectral theorem 71929.1 Operator -algebras, representations and GNS construction 71929.2 Spectral theorem, spectral measures, projection valued measures,

functional calculus 723

30 Rened algebraic quantisation (RAQ) and direct integraldecomposition (DID) 729

30.1 RAQ 72930.2 Master Constraint Programme (MCP) and DID 735

31 Basics of harmonic analysis on compact Lie groups 74631.1 Representations and Haar measures 74631.2 The Peter and Weyl theorem 752

32 Spin-network functions for SU(2) 75532.1 Basics of the representation theory of SU(2) 75532.2 Spin-network functions and recoupling theory 75732.3 Action of holonomy operators on spin-network functions 76232.4 Examples of coherent state calculations 765

33 + Functional analytic description of classical connectiondynamics 770

33.1 Innite-dimensional (symplectic) manifolds 770

References 775Index 809

Over half a century of collective study has not diminished the fascination ofsearching for a consistent theory of quantum gravity. I rst encountered thesubject in 1969 when, as a young researcher, I spent a year in Trieste work-ing with Abdus Salam who, for a while, was very interested in the subject. Inthose days, the technical approaches adopted for quantum gravity depended verymuch on the background of the researcher: those, like myself, from a theoreti-cal particle-physics background used perturbative quantum eld theory; thosewhose background was in general relativity tended to use relatively elementaryquantum theory, but taking full account of the background general relativity(which the other scheme did not).

The perturbative quantum eld theory schemes foundered on intractable ultra-violet divergences and gave way to super-gravity the super-symmetric exten-sion of standard general relativity. In spite of initial optimism, this approachsuccumbed to the same disease and was eventually replaced by the far moreambitious superstring theories. Superstring theory is now the dominant quan-tum gravity programe in terms of the number of personnel involved and thenumber of published papers, per year, per unit researcher.

However, notwithstanding my early training as a quantum eld theorist, Iquickly became fascinated by the canonical quantization, or quantum geome-try, schemes favored by those coming from general relativity. The early attemptsfor quantizing the metric variables were rather nave, and took on various formsaccording to how the intrinsic constraints of classical general relativity are han-dled. In the most popular approach, the constraints are imposed on the statevectors and give rise to the famous WheelerDeWitt equation arguably one ofthe most elegant equations in theoretical physics, and certainly one of the mostmathematically ill-dened. Indeed, it was the very intractability of this equationthat rst intrigued me and prompted me to see what could be done with moresophisticated quantization methods. After much eort it became clear that theanswer was not much.

The enormous diculty of the canonical quantum gravity scheme eventuallycaused it to go into something of a decline, until new life was imparted withAshtekars discovery of a set of variables in which the constraint equations sim-plify signicantly. This scheme slowly morphed into loop quantum gravity: anapproach which has, for the rst time, allowed real insight into what a non-perturbative quantisation of general relativity might look like. A number of

xviii Foreword

genuine results were obtained, but it became slowly apparent that the old prob-lems with the WheelerDeWitt equation were still there in transmuted form,and the critical Hamiltonian constraint was still ill-dened.

It was at this point that Thomas Thiemann the author of this book enteredthe scene. I can still remember the shock I felt when I rst read the papers heput onto the web dealing with the Hamiltonian constraint. Suddenly, someonewith a top-rate mathematical knowledge had addressed this critical questionanew, and with considerable success. Indeed, Thiemann succeeded with loopquantum gravity where I had failed with the old WheelerDeWitt equation, andhe has gone on since that time to become one of the internationally acknowledgedexperts in loop quantum gravity.

Thiemanns deep knowledge of mathematics applied to quantum gravity isevident from the rst page of this magnicent book. The subject is exploredin considerable generality and with real mathematical depth. The author startsfrom rst principles with a general introduction to quantum gravity, and thenproceeds to give, what is by far, the most comprehensive, and mathematicallyprecise, exposition of loop quantum gravity that is available in the literature. Thereader should be warned though that, when it comes to mathematics, the authortakes no hostages, and a good knowledge of functional analysis and dierentialgeometry is assumed from the outset. Still, that is how the subject is these days,and anyone who seriously aspires to work in loop quantum gravity would beadvised to gain a good knowledge of this type of mathematics. In that sense,this is a text that is written for advanced graduate students, or professionalswho work in the area.

My graduate students not infrequently ask me what I think of the currentstatus of canonical quantum gravity and, in particular, what I think the chancesare of ever making proper mathematical sense of the constraints that dene thetheory. For some years now I have replied to the eect that, if anybody can doit, it will be Thomas Thiemann and, if he cannot do it, then probably nobodywill. Anyone who reads right through this major new work will understand whyI place so much trust in the authors ability to crack this central problem ofquantum gravity.

Chris Isham,Professor of Theoretical Physics at

The Blankett Laboratory, Imperial College, London

Quantum General Relativity (QGR) or Quantum Gravity for short is, by def-inition, a Quantum (Field) Theory of Einsteins geometrical interpretation ofgravity which he himself called General Relativity (GR). It is a theory whichsynthesises the two fundamental building blocks of modern physics, that is, (1)the generally relativistic principle of background independence, sometimes calledgeneral covariance and (2) the uncertainty principle of quantum mechanics.

The search for a viable QGR theory is almost as old as Quantum Mechan-ics and GR themselves, however, despite an enormous eort of work by a vastamount of physicists over the past 70 years, we still do not have a credible QGRtheory. Since the problem is so hard, QGR is sometimes called the holy grail ofphysics. Indeed, it is to be expected that the discovery of a QGR theory revolu-tionises our current understanding of nature in a way as radical as both GeneralRelativity and Quantum Mechanics did.

What we do have today are candidate theories which display some promisingfeatures that one intuitively expects from a quantum theory of gravity. They areso far candidates only because for each of them one still has to show, at the endof the construction of the theory, that it reduces to the presently known standardmodel of matter and classical General Relativity at low energies, which is theminimal test that any QGR theory must pass.

One of these candidates is Loop Quantum Gravity (LQG). LQG is a modernversion of the canonical or Hamiltonian approach to Quantum Gravity, originallyintroduced by Dirac, Bergmann, Komar, Wheeler, DeWitt, Arnowitt, Deser andMisner. It is modern in the sense that the theory is formulated in terms ofconnections (gauge potentials) rather than metrics. It is due to this fact thatthe theory was called Loop Quantum Gravity since theories of connections arenaturally described in terms of Wilson loops. This also brings GR much closer tothe formulation of the other three forces of nature, each of which is described interms of connections of a particular YangMills theory for which viable quantumtheories exist. Consequently, the connection reformulation has resulted in rapidprogress over the past 20 years.

The purpose of this book is to provide a self-contained treatise on canon-ical and in particular Loop Quantum Gravity. Although the theory is stillunder rapid development and the present book therefore is at best a snap-shot, the eld has now matured enough in order to justify the publication ofa new textbook. The literature on LQG now comprises more than a thousand

Preface

articles scattered over a vast number of journals, reviews, proceedings and con-ference reports. Structures which were believed to be essential initially turnedout to be negligible later on and vice versa, thus making it very hard for thebeginner to get an overview of the subject. We hope that this book serves as ageodesic through the literature enabling the reader to move quickly from thebasics to the frontiers of current research. By denition, a geodesic cannot touchon all the subjects of the theory and we apologise herewith to our colleaguesif we were unable to cover their work in this single volume manuscript. How-ever, guides to further reading and a detailed bibliography try to compensatefor this incompleteness. A complete listing of all LQG-related papers, which isperiodically being updated, can be found in [1,2].1

Loop Quantum Gravity is an attempt to construct a mathematically rigorous,background-independent, non-perturbative Quantum Field Theory of LorentzianGeneral Relativity and all known matter in four spacetime dimensions, notmore and not less. In particular, no claim is made that LQG is a unied the-ory of everything predicting, among other things, matter content and dimen-sionality of the world. Hence, currently there is no restriction on the allowedmatter couplings although these might still come in at a later stage whenderiving the low energy limit. While the connection formulation works onlyin four spacetime dimensions and in that sense is a prediction, higherp-formformulations in higher dimensions are conceivable. Matter and geometry arenot unied in the sense that they are components of one and the same geo-metrical object, however, they are unied under the four-dimensional dieo-morphism group which in perturbative approaches is broken. LQG provides auniversal framework for how to combine quantum theory and General Relativ-ity for all possible matter and in that sense is robust against the very likelydiscovery of further substructure of matter between the energy scales of theLHC and the Planck scale which dier by 16 orders of magnitude. This is almostthe same number of orders of magnitude as between 1 mm and the length scalesthat the LHC can resolve, and we found a huge amount of substructure there.

The stress on mathematical rigour is here no luxurious extra baggage buta necessity: in a eld where, to date, no experimental input is available,mathematical consistency is the only guiding principle to construct the theory.The strategy is to combine the presently known physical principles and todrive them to their logical frontiers without assuming any extra, unobservedstructure such as extra dimensions and extra particles. This deliberatelyconservative approach has the advantage of either producing a viable theory orof deriving which extra structures are needed in order to produce a successfultheory. Indeed, it is conceivable that at some point in the development of thetheory a quantum leap is necessary, similar to Heisenbergs discovery that the

1 See also the URLs http://www.nucleares.unam.mx/corichi/lqgbib.pdf andhttp://www.matmor.unam.mx/corichi/lqgbib.pdf.

Preface xxi

BohrSommerfeld quantisation rules can be interpreted in terms of operators.The requirement to preserve background independence has already led to new,fascinating mathematical structures. For instance, a fundamental discreteness ofspacetime at the Planck scale of 1033 cm seems to be a prediction of the theorywhich is a rst substantial evidence for a theory in which the gravitational eldacts as a natural cuto of the usual ultraviolet divergences of QFT.

Accordingly, the present text tries to be mathematically precise. We willdevelop in depth the conceptual and mathematical framework underlying LQG,stating exact denitions and theorems including complete proofs. Many of thecalculations or arguments used during the proofs cannot be found anywhere inthe literature detailed as they are displayed here. We have supplied a vast amountof mathematical background information so that the book can be read by readerswith only basic prior knowledge of GR and QFT without having to consult toomuch additional literature. We have made an eort to stress the basic principlesof canonical QGR, of which LQG is just one possible incarnation based on aspecic choice of variables.

For readers who want to get acquainted rst with the physical ideas and con-ceptual aspects of LQG before going into mathematical details, we strongly rec-ommend the book by Carlo Rovelli [3]. The two books are complementary in thesense that they can be regarded almost as Volume I (Introduction and Concep-tual Framework) and Volume II (Mathematical Framework and Applications)of a general presentation of QGR in general and LQG in particular. While thisbook also develops a tight conceptual framework, the book by Carlo Rovelli ismuch broader in that aspect. Recent review articles can be found in [414]. Thestatus of the theory a decade ago is summarised in the books [1517].

The present text is aimed at all readers who want to nd out in detail howLQG works, conceptually and technically, enabling them to quickly develop theirown research on the subject. For instance, the author taught most of the materialof this book in a two-semester course to German students in physics and mathe-matics who were in their sixth semester of diploma studies or higher. After thatthey could complete diploma theses or PhD theses on the subject without muchfurther guidance. Unfortunately, due to reasons of space, exercises and their solu-tions had to be abandoned from the book, see [12] for a selection. We hope toincorporate them in an extended future edition. As we have pointed out, LQG isfar from being a completed theory and aspects of LQG which are at the frontierof current research and whose details are still under construction will be criticallydiscussed. This will help readers to get an impression of what important openproblems there are and hopefully encourage them to address these in their ownresearch.

The numerous suggestions for improvements to the previous online versionof this book (http://www.arxiv.org/list/gr-qc/0110034) by countless colleaguesis gratefully acknowledged, in particular those by Jurgen Ehlers, ChristianFleischhack, Stefan Hofmann, Chris Isham, Jurek Lewandowski, Robert Oeckl,Hendryk Pfeier, Carlo Rovelli, Hanno Sahlmann and Oliver Winkler. Special

xxii Preface

thanks go to my students Johannes Brunnemann, Bianca Dittrich and KristinaGiesel for a careful reading of the manuscript and especially to Kristina Gieselfor her help with the gures.

Posvwa svoe ene Tatne.Ebenso gewidmet meinen Sohnen Andreas und Maximilian.

Thomas ThiemannBerlin, Toronto 20012007

Symbol Meaning

G = 6.671011 m3 kg1 s2

Newtons constant

= 16G/c3 gravitational coupling constantp =

1033 cm Planck length

mp =

//c 1019 GeV/c2

Planck mass

Q YangMills coupling constantM, dim(M) = D + 1 spacetime manifold, dim() = D abstract spatial manifold spatial manifold embedded into MG compact gauge groupLie(G) Lie algebraN 1 rank of gauge group, , , .. = 0, 1, . . . , D tensorial spacetime indicesa, b, c, .. = 1, . . . , D tensorial spatial indicesa1..aD LeviCivita totally skew tensor pseudo density

of weight 1g spacetime metric tensorqab spatial (intrinsic) metric tensor of Kab extrinsic curvature of R curvature tensorh group elements for general Ghmn, m, n, o, .. = 1, . . . , N matrix elements for general GI, J,K, .. = 1, 2, . . . ,dim (G) Lie algebra indices for general G I Lie algebra generators for general GkIJ = tr( IJ)/N := IJ : CartanKilling metric

for G[ I , J ] = 2fIJ

KK structure constants for G(h) (irreducible) representations for general G or

algebrah group elements for SU(2)hAB , A,B,C, .. = 1, 2 matrix elements for SU(2)i, j, k, .. = 1, 2, 3 Lie algebra indices for SU(2)i Lie algebra generators for SU(2)

xxiv Notation and conventions

kij = ij CartanKilling metric for SU(2)fij

k = ijk structure constants for SU(2)j(h) (irreducible) representations for SU(2) with

spin jA connection on G-bundle over AIa pull-back of A to by local sectionA, oA; A = 1, 2 AA := AB AB = 1: spinor dyadA , oA ; A = 1, 2 primed (complex coinjugate) spinor dyadg gauge transformation or element of

complexication of GP principal G-bundleA connection on SU(2)-bundle over Aia pull-back of A to by local sectionE pseudo-(D 1)-form in vector bundle

associated to G-bundle under adjointrepresentation

EIa1..,aD1 := kIJa1..,aDEaDJ : pull-back of E to bylocal section

E pseudo-(D 1)-form in vector bundleassociated to SU(2)-bundle under adjointrepresentation

Eia1..,aD1 := kija1..,aDEaDj : pull-back of E to bylocal section

Eaj := a1..aD1(E)ka1..aD1kjk/((D 1)!):

electric eldse one-form co-vector bundle associated to the

SU(2)-bundle under the dening representation(D-bein)

eia pull-back of e to by local sectionia pull-back by local section of SU(2) spin

connection over R,X right-invariant vector eld on GL Left-invariant vector eld on GY = iX momentum vector eldM phase spaceE Banach manifold or space of smooth electric

eldsT(a1..an) :=

1n!

Sn Ta(1)..a(n) : symmetrisation of

indicesT[a1..an] :=

1n!

Sn sgn() Ta(1)..a(n) :

antisymmetrisation of indicesA space of smooth connectionsG space of smooth gauge transformations

Notation and conventions xxv

A space of distributional connectionsAC space of smooth complex connectionsGC space of smooth complex gauge transformationsG space of distributional gauge transformationsA/G space of smooth connections modulo smooth

gauge transformationsA/G space of distributional connections modulo

distributional gauge transformationsA/G space of distributional gauge equivalence classes

of connectionsAC space of distributional complex connectionsA/GC space of distributional complex gauge

equivalence classes of connectionsC set of semianalytic curves or classical

conguration spaceC quantum conguration spaceP set (groupoid) of semianalytic paths or set of

puncturesQ set (group) of semianalytic closed and

basepointed pathsL set of tame subgroupoids of P or general label

setS set of tame subgroups of Q (hoop group) or set

of spin-network labelsl subgroupoids spin-net= spin-network label[s] (singular) knot-net= dieomorphism

equivalence class of s0 set of semianalytic, compactly supported graphs set of semianalytic, countably innite graphsDi() group of smooth dieomorphisms of Disa() group of semianalytic dieomorphisms of Disa,0() group of semianalytic dieomorphisms of

connected to the identityDi0 () group of analytic dieomorphisms of

connected to the identityDi() group of analytic dieomorphisms of (semi-)analytic dieomorphismc semianalytic curvep semianalytic pathe entire semianalytic path (edge) entire semianalytic closed path (hoop) or

algebra automorphism

xxvi Notation and conventions

semianalytic graphv vertex of a graphE() set of edges of V () set of vertexes of hp(A) = A(p) holonomy of A along p abstract partial order vector state or symplectic structure or

curvature two-formF pull-back to of 2 by a local section general state on algebraX, X, Y measure space or topological spaceL(X,Y ), L(X) linear (un)bounded operators between X,Y or

on XB(X,Y ), B(X) bounded operators between X,Y or on XK(X) compact operators on XB1(X) trace class operators on XB2(X) HilbertSchmidt operators on XB -algebra, , measureH general Hilbert spaceCyl space of cylindrical functionsD dense subspace of H equipped with a stronger

topologyD topological dual of DD algebraic dual of DH0 = L2(A, d0) uniform measure L2 spaceH innite tensor product extension of H0Cyll restriction of Cyl to functions cylindrical over l[.], (.) equivalence classesA,B abstract (-)algebra or C-algebra(A) spectrum on Abelian C-algebra character (maximal ideal) of unital Banach

algebra or group or characteristic function of aset

I, J ideal in abstract algebraP classical Poisson-algebraG automorphism group (of principal bre bundle)D Dirac or hypersurface deformation algebraM Master Constraint algebraM Master Constraint

In the rst section of this chapter we explain why the problem of quantum gravitycannot be ignored in present-day physics, even though the available acceleratorenergies lie way beyond the Planck scale. Then we dene what a quantum theoryof gravity and all interactions is widely expected to achieve and point out the twomain directions of research divided into the perturbative and non-perturbativeapproaches. In the third section we describe these approaches in more detail andnally in the fourth motivate our choice of canonical quantum general relativityas opposed to other approaches.

Why quantum gravity in the twenty-rst century?It is often argued that quantum gravity is not relevant for the physics of this cen-tury because in our most powerful accelerator, the LHC to be working in 2007,we obtain energies of the order of a few 103 GeV while the energy scale at whichquantum gravity is believed to become important is the Planck energy of 1019GeV. While that is true, it is false that nature does not equip us with particlesof energies much beyond the TeV scale; we have already observed astrophysicalparticles with energy of up to 1013 GeV, only six orders of magnitude away fromthe Planck scale. It thus makes sense to erect future particle microscopes not onthe surface of the Earth any more, but in its orbit. As we will sketch in this book,even with TeV energy scales one might speculate about quantum gravity eectsin the close future with -ray burst physics and the GLAST detector. Next,quantum gravity eects in the early universe might have left their ngerprintin the cosmological microwave background radiation (CMBR) and new satellitessuch as WMAP and PLANCK which have considerably increased the precision ofexperimental cosmology might reveal those. Notice that these data have alreadygiven us new cosmological puzzles recently, namely they have, for the rst time,enabled us to reliably measure the energy budget of the universe: about 70%is a so-called dark energy component which could be a positive1 cosmologicalconstant, about 25% is a dark matter component which is commonly believedto be due to a weakly interacting massive particle (WIMP) (possibly supersym-metric) and only about 5% is made out of baryonic matter. Here dark means

1 Recent independent observations all indicate that the expansion of the universe is currentlyaccelerating.

2 Introduction: Dening quantum gravity

that these unknown forms of matter do not radiate, they are invisible. Hence wesee that we only understand 5% of the matter in the universe and at least asfar as dark energy is concerned, quantum gravity could have a lot to do with it.What we want to argue here is that quantum gravity is not at all of academicinterest but possibly touches on brand new observational data which point atnew physics beyond the standard model and are of extreme current interest. See,for example, [1820] for recent accounts of modern cosmology.

But even apart from these purely experimental considerations, there are goodtheoretical reasons for studying quantum gravity. To see why, let us summariseour current understanding of the fundamental interactions:

Embarassingly, the only quantum elds that we fully understand to date infour dimensions are free quantum elds on four-dimensional Minkowski space.Formulated more provocatively:

In four dimensions we only understand an (innite) collection ofuncoupled harmonic oscillators on Minkowski space!

In order to leave the domain of these rather trivial and unphysical (since non-interacting) quantum eld theories, physicists have developed two techniques:perturbation theory and quantum eld theory on curved backgrounds. Thismeans the following: with respect to accelerator experiments, the most importantprocesses are scattering amplitudes between particles. One can formally writedown a unitary operator that accounts for the scattering interaction betweenparticles and which maps between the well-understood free quantum eld Hilbertspaces in the far past and future. Famously, by Haags theorem [21] wheneverthat operator is really unitary, there is no interaction and if it is not unitary,then it is ill-dened giving rise to the ultraviolet divergences of ordinary QFT. Infact, one can only dene the operator perturbatively by writing down the formalpower expansion in terms of the generator of the would-be unitary transforma-tion between the free quantum eld theory Hilbert spaces. The resulting series isdivergent order by order but if the theory is renormalisable then one can makethese orders articially nite by a regularisation and renormalisation procedurewith, however, no control on convergence of the resulting series. Despite thesedrawbacks, this recipe has worked very well so far, at least for the electroweakinteraction.

Until now, all we have said applies only to free (or perturbatively interact-ing) quantum elds on Minkowski spacetime for which the so-called Wightmanaxioms [21] can be veried. Let us summarise them for the case of a scalar eldin (D + 1)-dimensional Minkowski space:

W1 RepresentationThere exists a unitary and continuous representation U : P B(H) of thePoincare group P on a Hilbert space H.

Why quantum gravity in the twenty-rst century? 3

W2 Spectral conditionThe momentum operators P have spectrum in the forward lightcone:P

P 0; P 0 0.W3 Vacuum

There is a unique Poincare invariant vacuum state U(p) = for allp P .

W4 CovarianceConsider the smeared eld operator-valued tempered distributions (f) =RD+1

dD+1x(x)f(x) where f S(RD+1) is a test function of rapiddecrease. Then nite linear combinations of the form (f1) . . . (fN ) liedense in H (that is, is a cyclic vector) and U(p)(f)U(p)1 = (f p)for any p P .

W5 Locality (causality)Suppose that the supports (the set of points where a function is dier-ent from zero) of f, f are spacelike separated (that is, the points oftheir supports cannot be connected by a non-spacelike curve) then [(f),(f )] = 0.

The most important objects in this list are those that are highlighted inboldface letters: the xed, non-dynamical Minkowski background metric withits well-dened causal structure, its Poincare symmetry group P , the associatedrepresentation U(p) of its elements, the invariant vacuum state and nallythe xed, non-dynamical topological, dierentiable manifold RD+1. Thusthe Wightman axioms assume the existence of a non-dynamical, Minkowskibackground metric which implies that we have a preferred notion of causality(or locality) and its symmetry group, the Poincare group from which one buildsthe usual Fock Hilbert spaces of the free elds. We see that the whole structureof the theory is heavily based on the existence of these objects which come witha xed, non-dynamical background metric on a xed, non-dynamical topologicaland dierentiable manifold.

For a general background spacetime, things are already under much lesscontrol: we still have a notion of causality (locality) but generically no symmetrygroup any longer and thus there is no obvious generalisation of the Wightmanaxioms and no natural perturbative Fock Hilbert space any longer. These obsta-cles can partly be overcome by the methods of algebraic quantum eld theory [22]and the so-called microlocal analysis [2326] (in which the locality axiom is takencare of pointwise rather than globally), which recently have also been employedto develop perturbation theory on arbitrary background spacetimes [2733] byinvoking the mathematically more rigorous implementation of the renormal-isation programme developed by Epstein and Glaser in which no divergentexpressions ever appear at least order by order (see, e.g., [34]). This way onemanages to construct the interacting elds, at least perturbatively, on arbitrarybackgrounds.

Introduction: Dening quantum gravity

In order to go beyond a xed background one can consider all backgroundssimultaneously [35, 36]. Namely, the notion of a local quantum eld theoryA(M, g) (thought of as a unital C-algebra for convenience) on a given curvedbackground spacetime (M, g) can be generalised in the following way:2 givenan isometric embedding : (M, g) (M , g) of one spacetime into another,one relates A(M, g), A(M , g) by asking that there is a -algebraic homomor-phism : A(M, g) A(M , g). The homomorphisms could for instancejust act geometrically by pulling back the elds. More abstractly, what onehas then is the category Man whose objects are globally hyperbolic spacetimes(M, g) and whose morphisms are isometric embeddings with unit 1(M,g) := idM ,the identity dieomorphism. On the other hand, we have the category Algwhose objects are unital C-algebras A and whose morphisms are injective-homomorphisms with unit 1A = idA, the identity element in the algebra.A local quantum eld is then a covariant functor A : Man Alg; (M, g) A(M, g), which relates objects and morphisms of Man with those ofAlg. The functor is called causal if those quantum eld theories A(Mj , gj)for which there exist isometric embeddings j : (Mj , gj) (M, g); j = 1, 2 sothat 1(M1), 2(M2) are spacelike separated with respect to g satisfy thecausality axiom [1(A(M1, g1)), 2(A(M2, g2))] = { 0} . The functor is said toobey the time slice axiom when (A(M, g)) = A(M , g)) for all isometries : (M, g) (M , g) such that (M) contains a Cauchy surface for (M , g).This framework is background-independent because the functor A considers allbackgrounds (M, g) simultaneously.

Unfortunately, QFT on curved spacetimes, even stated in this background-independent way, is only an approximation to the real world because it com-pletely neglects the backreaction between matter and geometry which classi-cally is expressed in Einsteins equations. Moreover, it neglects the fact thatthe gravitational eld must be quantised as well, as we will argue below. Onecan try to rescue the framework of ordinary QFT by studying the quantumexcitations around a given classical background metric, possibly generalised inthe above background-independent way. However, not only does this result in anon-renormalisable theory without predictive power when treating the gravita-tional eld in the same fashion, it is also unclear whether the procedure leadsto (unitarily) equivalent results when using backgrounds which are physicallydierent, such as two Schwarzschild spacetimes with dierent mass (the cor-responding spacetimes are not isometric). More seriously, it is expected thatespecially in extreme astrophysical or cosmological situations (black holes, bigbang) the notion of a classical, smooth spacetime breaks down altogether!In other words, the uctuations of the metric operator become deeply quantumand there is no semiclassical notion of a spacetime any more, similarly to the

2 The following paragraph can be skipped on a rst reading, however, the appearing notionsare all explained in this book (see, e.g., Denition 6.2.6 and Chapter 29).

Why quantum gravity in the twenty-rst century? 5

energy spectrum of the hydrogen atom far away from the continuum limit. It isprecisely here where a full-edged quantum theory of gravity is needed: we mustbe able to treat all backgrounds on a common footing, otherwise we will neverunderstand what really happens in a Hawking process when a black hole losesmass due to radiation. Moreover, we need a background-independent theory ofGR where the lightcones themselves start uctuating and hence locality becomesa fuzzy notion. Let us phrase this again, provocatively, as:

The whole framework of ordinary quantum eld theory breaks downonce we make the gravitational eld (and the dierentiable manifold)dynamical, once there is no background metric any longer!

Combining these issues, one can say that we have a working understanding ofscattering processes between elementary particles in arbitrary spacetimes as longas the backreaction of matter on geometry can be neglected and that the cou-pling constant between non-gravitational interactions is small enough (with QCDbeing an important exception) since then the classical Einstein equation, whichsays that curvature of geometry is proportional to the stress energy of matter,can be approximately solved by neglecting matter altogether. Thus, in this limit,it seems fully sucient to have only a classical theory of general relativity andperturbative quantum eld theory on curved spacetimes.

From a fundamental point of view, however, this state of aairs is unsatisfac-tory for many reasons among which we have the following:

(i) Classical geometry quantum matter inconsistencyThere are two kinds of problem with the idea of keeping geometry classicalwhile matter is quantum:(i1) Backreaction

At a fundamental level, the backreaction of matter on geometry cannotbe neglected. Namely, geometry couples to matter through Einsteinsequations

R 12R g = T [g]

and since matter underlies the rules of quantum mechanics, the right-hand side of this equation, the stressenergy tensor T [g], becomesan operator. One has tried to keep geometry classical while matter isquantum mechanical by replacing T [g] by the Minkowski vacuum expectation value < , T [] >, but the solution of this equationwill give g = which one then has to feed back into the denitionof the vacuum expectation value, and so on. Notice that the notionof vacuum itself depends on the background metric, so that this is ahighly non-trivial iteration process. The resulting iteration does not

6 Introduction: Dening quantum gravity

converge in general [37]. Thus, such a procedure is also inconsistent,whence we must quantise the gravitational eld as well. This leads tothe quantum Einstein equations

R 12R g = T [g]

Of course, this equation is only formal at this point and must beembedded into an appropriate Hilbert space context.

(i2) UV regimeThere is another piece of evidence for the need to quantise geometry:recall that in perturbative QFT one integrates over virtual particlesin higher loop diagrams with arbitrarily large energy. Suppose thatsuch a particle has energy E and momentum P E/c in some restframe. According to quantum mechanics, such a particle has a lifetime h/E and a spatial extension given by the Compton radius hc/E. According to classical GR, such a lump of energy collapses to ablack hole if the Compton radius drops below the Schwarzschild radiusr GE/c4, in other words, when the energy exceeds the Planck energyEp =

hc/Gc2. The problem is now not only that in ordinary QFT this

general relativistic eect is neglected, but moreover that this eect leadsto new processes: according to the Hawking eect, after the lifetime the black hole evaporates. However, it evaporates into particles of allpossible species. Suppose for instance that the original particle was aneutrino. All that the resulting black hole remembers is its mass andspin. Now while the neutrino only interacts electroweakly according tothe standard model, the black hole can produce gluons and quarks,which is impossible within the standard model.

Of course, all of these arguments are only heuristic, however, they revealthat it is problematic to combine classical geometry with quantum matter.They suggest that it is problematic or even inconsistent to resolve spacetimedistances below the Planck scale p =

hcG/c2. It is due to considerations

of this kind that one expects that gravity provides a natural UV cuto forQFT. If that is the case, then it is natural to expect that the quantumspacetime structure reveals a discrete structure at Planck scale. We will seea particular incarnation of this idea in LQG.

(ii) Inherent classical geometry inconsistencyEven without quantum theory at all Einsteins eld equations predict space-time singularities (black holes, big bang singularities, etc.) at which theequations become meaningless. In a truly fundamental theory, there is noroom for such breakdowns and it is suspected by many that the theory curesitself upon quantisation in analogy to the hydrogen atom whose stability isclassically a miracle (the electron should fall into the nucleus after a nite

Why quantum gravity in the twenty-rst century? 7

time lapse due to emission of Bremsstrahlung) but is easily explained byquantum theory which bounds the electrons energy from below.

(iii) Inherent quantum matter inconsistencyAs outlined above, perturbative quantum eld theory on curved spacetimesis itself also ill-dened due to its UV (short distance) singularities whichcan be cured only with an ad hoc recipe order by order which lacks afundamental explanation; moreover, the perturbation series is usually diver-gent. Besides that, the corresponding innite vacuum energies being usuallyneglected in such a procedure contribute to the cosmological constant andshould have a large gravitational backreaction eect. That such energy sub-tractions are quite signicant is maybe best demonstrated by the Casimireect. Now, since general relativity possesses a fundamental length scale,the Planck length p 1033 cm, it has been argued ever since that grav-itation plus matter should give a nite quantum theory since gravitationprovides the necessary, built-in, short distance cuto.

(iv) Cosmological constant problemHowever, that cuto cannot work naively: consider for simplicity a free mass-less scalar eld on Minkowski space. The dierence between the Hamiltonianand its normal ordered version is given by the divergent expression

H : H := h

d3x[(x, y)]y=x = h

d3x

d3k |k|

where is the at space Laplacian. If we assume a naive momentum cut-o due to quantum gravity at |k| 1/P the divergent momentum integralbecomes proportional to 4P . Comparing this with the cosmological con-stant Hamiltonian G

d3x

det(q) where is the cosmological constant, G

is Newtons constant and q is the spatial metric (which is at on Minkowskispace) then we conclude that 2P 1 where hG = 2P was used. However,experimentally we nd 2P 10120. Thus the cosmological constant isunnaturally small and presents the worst ne-tuning problem ever encoun-tered in physics. Notice that the cosmological constant is a possible candi-date for dark energy.

(v) Perturbative quantum gravity inconsistencyGiven the fact that perturbation theory works reasonably well if the couplingconstant is small for the non-gravitational interactions on a backgroundmetric it is natural to try whether the methods of quantum eld theoryon curved spacetime work as well for the gravitational eld. Roughly, theprocedure is to write the dynamical metric tensor as g = + h where isthe Minkowski metric and h is the deviation of g from it (the graviton) andthen to expand the Lagrangian as an innite power series in h. One arrivesat a formal, innite series with nite radius of convergence which becomesmeaningless if the uctuations are large. Although the naive power countingargument implies that general relativity so dened is a non-renormalisable

8 Introduction: Dening quantum gravity

theory, it was hoped that due to cancellations of divergences the perturba-tion theory could actually be nite. However, that this hope was unjustiedwas shown in [38, 39] where calculations demonstrated the appearance ofdivergences at the two-loop level, which suggests that at every order ofperturbation theory one must introduce new coupling constants which theclassical theory did not know about and one loses predictability.

It is well known that the (locally) supersymmetric extension of a givennon-supersymmetric eld theory usually improves the ultraviolet conver-gence of the resulting theory as compared with the original one due tofermionic cancellations [40]. It was therefore natural to hope that quantisedsupergravity might be nite. However, in [41] a serious argument against theexpected cancellation of perturbative divergences was raised and recentlyeven the again popular (due to its M-theory context) most supersymmet-ric 11D last hope supergravity theory was shown not to have the magicalcancellation property [4244].

Summarising, although a denite proof is still missing up to date (mainlydue to the highly complicated algebraic structure of the Feynman rulesfor quantised supergravity) it is today widely believed that perturbativequantum eld theory approaches to quantum gravity are meaningless.

The upshot of these considerations is that our understanding of quantum eldtheory and therefore fundamental physics is quite limited unless one quantisesthe gravitational eld as well. Being very sharply critical one could say:

The current situation in fundamental physics can be compared withthe one at the end of the nineteenth century: while one had a success-ful theory of electromagnetism, one could not explain the stability ofatoms. One did not need to worry about this from a practical point ofview since atomic length scales could not be resolved at that time butfrom a fundamental point of view, Maxwells theory was incomplete.The discovery of the mechanism for this stability, quantum mechanics,revolutionised not only physics. Similarly, today we still have no thor-ough understanding for the stability of nature in the sense discussedabove and it is similarly expected that the more complete theory ofquantum gravity will radically change our view of the world. Thatis, considering the metric as a quantum operator will bring us beyondstandard model physics even without the discovery of new forces, par-ticles or extra dimensions.

The role of background independence

The twentieth century has dramatically changed our understanding of nature: itrevealed that physics is based on two profound principles, quantum mechanicsand general relativity. Both principles revolutionise two pivotal structures of

The role of background independence 9

Newtonian physics. First, the determinism of Newtons equations of motion evap-orates at a fundamental level, rather dynamics is reigned by probabilities under-lying the Heisenberg uncertainty obstruction. Second, the notion of absolute timeand space has to be corrected; space and time and distances between points ofthe spacetime manifold, that is, the metric, become themselves dynamical, geom-etry is no longer just an observer. The usual Minkowski metric ceases to be adistinguished, externally prescribed, background structure. Rather, the laws ofphysics are background-independent, mathematically expressed by the classicalEinstein equations which are generally (or four-dieomorphism) covariant. As wehave argued, it is this new element of background independence brought in withEinsteins theory of gravity which completely changes our present understandingof quantum eld theory.

A satisfactory physical theory must combine both of these fundamental prin-ciples, quantum mechanics and general relativity, in a consistent way and will becalled Quantum Gravity. However, the quantisation of the gravitational eldhas turned out to be one of the most challenging unsolved problems in theoreticaland mathematical physics. Although numerous proposals towards a quantisationhave been made since the birth of general relativity and quantum theory, noneof them can be called successful so far. This is in sharp contrast to what we seewith respect to the other three interactions whose description has culminatedin the so-called standard model of matter, in particular, the spectacular successof perturbative quantum electrodynamics whose theoretical predictions could beveried to all digits within the experimental error bars until today.

Today we do not have a theory of quantum gravity, what we have is:

1. The Standard Model, a quantum theory of the non-gravitational interactions(electromagnetic, weak and strong) or matter which, however, completelyignores General Relativity.

2. Classical General Relativity or geometry, which is a background-independenttheory of all interactions but completely ignores quantum mechanics.

What is so special about the gravitational force that it has persisted in itsquantisation for about 70 years already? As outlined in the previous section, theanswer is simply that today we only know how to do QFT on xed backgroundmetrics. The whole formalism of ordinary QFT relies heavily on this backgroundstructure and collapses to nothing when it is missing. It is already much moredicult to formulate a QFT on a non-Minkowski (curved) background but itseems to become a completely hopeless task when the metric is a dynamical,even uctuating quantum eld itself. This underlines once more the source ofour current problem of quantising gravity: we have to learn how to do QFT ona dierential manifold (or something even more rudimentary, not even relyingon a xed topological, dierentiable manifold) rather than a spacetime.

In order to proceed, today a high-energy physicist has the choice betweenthe following two, extreme approaches. Either the particle physicists, whoprefers to take over the well-established mathematical machinery from QFT

10 Introduction: Dening quantum gravity

on a background at the price of dropping background independence altogetherto begin with and then tries to nd the true background-independent theoryby summing the perturbation series (summing over all possible backgrounds).Or the quantum geometers, who believes that background independence lies atthe heart of the solution to the problem and pays the price to have to inventmathematical tools that go beyond the framework of ordinary QFT right fromthe beginning. Both approaches try to unravel the truly deep features that areunique to Einsteins theory associated with background independence from dif-ferent ends.

The particle physicists language is perturbation theory, that is, one writesthe quantum metric operator as a sum consisting of a background piece and aperturbation piece around it, the graviton, thus obtaining a graviton QFT on aMinkowski background. We see that perturbation theory, by its very denition,breaks background independence and dieomorphism invariance at every niteorder of perturbation theory. Thus one can restore background independenceonly by summing up the entire perturbation series, which is of course not easy.Not surprisingly, as already mentioned, since h = 2p has negative mass dimen-sion in Planck units, applying this programme to Einsteins theory itself resultsin a mathematical disaster, a so-called non-renormalisable theory without anypredictive power. In order to employ perturbation theory, it seems that one hasto go to string theory which, however, requires the introduction of new additionalstructures that Einsteins classical theory did not know about: supersymmetry,extra dimensions and an innite tower of new and very heavy particles next tothe graviton. This is a fascinating but extremely drastic modication of generalrelativity and one must be careful not to be in conict with phenomenology assuperparticles, Kaluza Klein modes from the dimensional reduction and thoseheavy particles have not been observed until today. On the other hand, stringtheory has a good chance to be a unied theory of the perturbative aspects ofall interactions in the sense that all interactions follow from a common object,the string, thereby explaining the particle content of the world.

The quantum geometers language is a non-perturbative one, keeping back-ground independence as a guiding principle at every stage of the construction ofthe theory, resulting in mathematical structures drastically dierent from theones of ordinary QFT on a background metric. One takes Einsteins theoryabsolutely seriously, uses only the principles of General Relativity and quantummechanics and lets the theory build itself, driven by mathematical consistency.Any theory meeting these standards will be called Quantum General Relativity(QGR). Since QGR does not modify the matter content of the known interac-tions, QGR is therefore not in conict with phenomenology but also it does notobviously explain the particle content of the world. However, it tries to unify allinteractions in a dierent sense: all interactions must transform under a com-mon gauge group, the four-dimensional dieomorphism group which on the otherhand is almost completely broken in perturbative approaches.

Approaches to quantum gravity 11

Let us remark that even without specifying further details, any QGR theoryis a promising candidate for a theory that is free from two divergences of theso-called perturbation series of Feynman diagrams common to all perturbativeQFTs on a background metric: (1) each term in the series diverges due to theultraviolet (UV) divergences of the theory which one can cure for renormalis-able theories through so-called renormalisation techniques and (2) the seriesof these renormalised, nite terms diverges, one says the theory is not nite.The rst, UV, problem has a chance to be absent in a background-independenttheory for a simple but profound reason: in order to say that a momentumbecomes large one must refer to a background metric with respect to which it ismeasured, but there simply is no background metric in the theory. The second,convergence, problem of the series might be void as well since there are simplyno Feynman diagrams! Thus, the mere existence of a consistent background-independent quantum gravity theory could imply a nite quantum theory of allinteractions. Of course, a successful quantum gravity theory must recover all theresults that have been obtained by perturbative techniques and that have beenveried in experiments.

Approaches to quantum gravity

The aim of the previous section was to convince the reader that background inde-pendence is, maybe, the Key Feature of quantum gravity to be dealt with. Nomatter how one deals with this issue, whether one starts from a perturbative (=background-dependent) or from a non-perturbative (= background-independent)platform, one has to invent something drastically new in order to quantise thegravitational eld. Roughly speaking, if one wants to keep perturbative renor-malisability as a criterion for a meaningful theory, then one has to increase theamount of symmetries, resulting in superstring theory which hopefully has Gen-eral Relativity and the standard model as an eective low-energy limit. (Comparethe historically similar case of the non-renormalisable Fermi model of the weakinteraction with massive gauge bosons which was replaced by the more symmet-ric and renormalisable electroweak YangMills theory.) If one considers GeneralRelativity as a fundamental theory then one cannot introduce extra structure,one has to give up the renormalisability principle and instead has to invent anew mathematical framework which can deal with background independence.(Compare the historically similar case of the bizarre ether model based on theNewtonian notion of absolute spacetime which was abandoned by the specialrelativity principle.)

We will now explain these approaches in more detail.

1. Perturbative approach: string theoryThe only known consistent perturbative approach to quantum gravity is stringtheory which has good chances to be a theory that unies all interactions.String theory [45] is not a eld theory in the ordinary sense of the word.

12 Introduction: Dening quantum gravity

Originally, it was a two-dimensional eld theory of worldsheets embeddedinto a xed, D-dimensional pseudo-Riemannian manifold (M, g) of Lorentziansignature which is to be thought of as the spacetime of the physical world.The Lagrangian of the theory is a kind of non-linear -model Lagrangian forthe associated embedding variables X (and their supersymmetric partnersin case of the superstring). If one perturbs g(X) = + h(X) as above andkeeps only the lowest order in X one obtains a free eld theory in two dimen-sions which, however, is consistent (Lorentz covariant) only when D + 1 = 26(bosonic string) or D + 1 = 10 (superstring), respectively. Strings propagat-ing in those dimensions are called critical strings, non-critical strings existbut have so far not played a signicant role due to phenomenological reasons.Remarkably, the mass spectrum of the particle-like excitations of the closedworldsheet theory contains a massless spin-two particle which one interpretsas the graviton. Until recently, the superstring was favoured since only therewas it believed to be possible to get rid of an unstable tachyonic vacuum stateby the GSO projection. However, one recently also tries to construct stablebosonic string theories [46].

Moreover, if one incorporates the higher-order terms h(X) of the stringaction, sucient for one-loop corrections, into the associated path integralone nds a consistent quantum theory up to one loop only if the backgroundmetric satises the Einstein equations. These are the most powerful outcomesof the theory: although one started out with a xed background metric, thebackground is not arbitrary but has to satisfy the Einstein equations up tohigher loop corrections, indicating that the one-loop eective action for thelow-energy quantum eld theory in those D dimensions is Einsteins theoryplus corrections. Finally, only recently has it been shown [47] that at leastthe type II superstring theories are one- and two-loop and, possibly, to allorders, nite. String theorists therefore argue to have found candidates for aconsistent theory of quantum gravity with the additional advantage that theydo not contain any free parameters (like those of the standard model) exceptfor the string tension.

These facts are very impressive, however, some cautionary remarks areappropriate, see also the beautiful review [48]: Vacuum degeneracy

Dimension D + 1 = 10, 26 is not the dimension of everyday physics so thatone has to argue that the extra D 3 dimensions are tiny in the KaluzaKlein sense although nobody knows the mechanism responsible for thisspontaneous compactication. According to [49] there exist at least 104

consistent, distinct CalabiYau compactications (other compacticationssuch as toroidal ones seem to be inconsistent with phenomenology), eachof which has an order of 102 free, continuous parameters (moduli) like thevacuum expectation value of the Higgs eld in the standard model. For eachcompactication of each of the ve string theories in D = 10 dimensions

Approaches to quantum gravity 13

and for each choice of the moduli one obtains a distinct low-energy eectivetheory. This is clearly not what one expects from a theory that aims to unifyall the interactions, the 18 (or more for massive neutrinos) free, continuousparameters of the standard model have been replaced by 102 continuousplus at least 104 discrete ones.

This vacuum degeneracy problem is not cured by the M-theory inter-pretation of string theory but it is conceptually simplied if certain con-jectures are indeed correct: string theorists believe (bearing on an impres-sively huge number of successful checks) that so-called T (or target space)and S (or strongweak coupling) duality transformations between all thesestring theories exist, which suggests that we do not have 104 unrelated 102-dimensional moduli spaces but that rather these 102-dimensional manifoldsintersect in singular, lower-dimensional submanifolds corresponding to cer-tain singular moduli congurations. This typically happens when certainmasses vanish or certain couplings diverge or vanish (in string theory thecoupling is related to the vacuum expectation value of the dilaton eld).Crucial in this picture are so-called D-branes, higher-dimensional objectsadditional to strings which behave like solitons (magnetic monopoles) inthe electric description of a string theory and like fundamental objects(electric degrees of freedom) in the S-dual description of the same stringtheory, much like the electricmagnetic duality of Maxwell theory underwhich strong and weak coupling are exchanged. Further relations betweendierent string theories are obtained by compactifying them in one wayand decompactifying them in another way, called a T-duality transforma-tion. The resulting picture is that there exists only one theory which hasall these compactication limits just described, called M-theory. Curiously,M-theory is an 11D theory whose low energy limit is 11D supergravity andwhose weak coupling limit is type IIA superstring theory (obtained by oneof these singular limits since the size of the 11th compactied dimensionis related to the string coupling again). Since 11D supergravity is also thelow-energy limit of the 11D supermembrane, some string theorists inter-pret M-theory as the quantised 11D supermembrane (see, e.g., [50,51] andreferences therein).

Phenomenology matchUntil today, no conclusive proof exists that for any of the compacticationsdescribed above we obtain a low-energy eective theory which is experi-mentally consistent with the data that we have for the standard model [52],although one seems to get at least rather close. The challenge in string phe-nomenology is to consistently and spontaneously break supersymmetry inorder to get rid of the so far non-observed superpartners. There is alsoan innite tower of very massive (of the order of the Planck mass andhigher) excitations of the string, but these are too heavy to be observable.More interesting are the KaluzaKlein modes whose masses are inverse

14 Introduction: Dening quantum gravity

proportional to the compactication radii and which have recently givenrise to speculations about sub-mm-range gravitational forces [53], whichone must make consistent with observation also.

Fundamental descriptionEven before the M-theory revolution, string theory has always been a the-ory without Lagrangian description, S-matrix element computations havebeen guided by conformal invariance but there is no interaction Hamil-tonian, string theory is a rst-quantised theory. Second quantisation ofstring theory, called string eld theory [54], has so far not attracted asmuch attention as it possibly deserves. However, a fascinating possibilityis that the 11D supermembrane, and thus M-theory, is an already second-quantised theory [55].

Background dependenceAs mentioned above, string theory is best understood as a free 2D eldtheory propagating on a 10D Minkowski target space plus perturbative cor-rections for scattering matrix computations. This is a heavily background-dependent description, issues like the action of the 10D dieomorphismgroup, the fundamental symmetry of Einsteins action, or the probabil-ity amplitude for the quantum evolution of one background into anothercannot be questioned. Perturbative string theory, as far as quantum grav-ity is concerned, can describe graviton scattering in a background space-time, however, the most interesting problems near classical singularitiesrequire a non-perturbative description, such as the fundamental descrip-tion of Hawking radiation. As a rst step in that direction, recently stringyblack holes have been discussed [56]. Here one uses so-called BPS D-branecongurations which are so special that one can do a perturbative cal-culation and extend it to the non-perturbative regime since the resultsare protected against non-perturbative corrections due to supersymmetry.So far this works only for extremely charged, supersymmetric black holeswhich are astrophysically not very realistic. But still these developmentsare certainly a move in the right direction since they use for the rst timenon-perturbative ideas in a crucial way and have been celebrated as one ofthe triumphs of string theory.

The landscapeComing back to the D-branes mentioned above, these are surfaces onwhich open strings must end (D stands for Dirichlet boundary condi-tions). Since these D-branes are completely arbitrary and not constrainedby the theory, M-theory contains as many vacua as there are D-brane con-gurations (sometimes called charges or uxes), which of course have tobe gauge-invariant, in particular supersymmetric. This makes the num-ber of string vacua plain innite [57] and the number of physically rel-evant (e.g., consistent with cosmological observations, supersymmetry,topology and/or stable) vacua has been estimated to be of the order of

15

10100 10 500 [58,59] or even innite [60] depending on ones assumptions (allanalyses count compactication possibilities as well). Whether this numberis innite or just very large seems to be currently under debate, however,the number seems to be robustly above the 1080 particles contained inthe observable (causally connected, i.e., of Hubble radius size) universe.This number of vacua, called the landscape, is so vast that some stringtheorists [61] employ the anthropic principle in order to rescue predictabil-ity of string theory, which is not unproblematic [62]. From the point ofview of a background-independent theory which in some sense describes allbackground-dependent quantum eld or string theories (i.e., vacua) simul-taneously, the landscape could be an artifact of trying to describe quantumgravity by a collection of background-dependent theories which are not con-nected to each other while they should be. See [63] for more details.

AdS/CFT and cosmologyIn a celebrated paper [64, 65], Maldacena conjectured that string theoryon an anti-de Sitter (AdS) background can be described by a conformalquantum eld theory (CFT) 3 on the boundary of the AdS space. For anintroduction to CFT, see [66]. This is yet another duality conjecture ofstring theory whose most studied incarnation is string theory on anAdS5 S5 background andN = 4 SuperYangMills theory (SYM). The latter isnite order by order in perturbation theory. The AdS/CFT correspondencecan be considered as a concrete application of the holographic principle(see, e.g., [67]).

Unfortunately, so far this conjecture has mostly been checked at thelevel of the low-energy limit of string theory, that is, the correspondingsupergravity theory, while there has been recent progress [68] as far as theconformal eld theory side of the correspondence is concerned, based on thediscovery of certain integrability structures. Moreover, in a mathematicallyprecise formulation of the conjecture [6972] one can show by the methodsof algebraic QFT (local quantum physics) that if the theory in the bulk isdescribed by a local Lagrangian then the boundary theory is non-local andvice versa. There is no contradiction because the full low-energy eectiveaction of string theory is non-local (containing an innite tower of cor-rections), however, it then becomes hard to verify the conjecture just usingthe tree term.

3 That is, a QFT on D-dimensional Minkowski space whose underlying Lagrangian is notonly invariant under the Poincar e group ISO(1 ,D 1) but also under conformaltransformations. The resulting enlarged group is called the conformal group and itselements g satisfy g = 2 where is the Minkowski metric and is an arbitraryfunction. For isometries = 1, for non-trivial conformal transformations = 1. TheAdS/CFT correspondence or conjecture is based on the fact that the isometry groups on anAdS space in D + 1 spacetime dimensions, as well as the conformal group of Minkowskispace in D spacetime dimensions, have (locally) the structure of SO(2 ,D).

16 Introduction: Dening quantum gravity

Notice that current observations indicate that our universe is in a deSitter phase (positive cosmological constant). However, a de Sitter back-ground, in contrast to an anti-de Sitter background, does not have a positiveenergy supersymmetric extension of the de Sitter algebra (the analogue ofthe Poincare algebra). One way to see this is to note that in supersymmetrictheories the energy is always positive while de Sitter space does not admita global timelike Killing eld and hence no positive energy. String theoriesbased on de Sitter space, if they exist, thus tend to be unstable since thecorresponding low-energy supergravity theories are. In general it is hard toformulate string theory on time-dependent backgrounds which, however,are the most relevant ones for cosmology. Quite generally it is simply nottrue that every solution of Einsteins equations without RaritaSchwingerelds has a supersymmetric extension including RaritaSchwinger elds,that is, not every Einstein space is compatible with supergravity (localsupersymmetry).

2. Non-perturbative approachesThe non-perturbative approaches to quantum gravity can be grouped into thefollowing ve main categories.2a. Canonical Quantum General Relativity

If one wanted to give a denition of this theory then one could say thefollowing:

Canonical Quantum General Relativity is an attemptto construct a mathematically rigorous, non-perturbative,background-independent Quantum Field Theory of four-dimensional, Lorentzian general relativity plus all knownmatter in the continuum.

This is the oldest approach and goes back to the pioneering workby Dirac [7376] started in the 1940s and was further developed byBergmann and Komar [7780] as well as Arnowittt, Deser and Mis-ner [81] in the 1950s and especially by Wheeler and DeWitt [8285] inthe 1960s. The idea of this approach is to apply the Legendre transformto the EinsteinHilbert action by splitting spacetime into space and timeand to cast it into Hamiltonian form. The resulting Hamiltonian His actually a so-called Hamiltonian constraint, that is, a Hamiltoniandensity which is constrained to vanish by the equations of motion. AHamiltonian constraint must occur in any theory that, like generalrelativity, is invariant under local reparametrisations of time. Accordingto Diracs theory of the quantisation of constrained Hamiltonian systems,one is now supposed to impose the vanishing of the quantisation H ofthe Hamiltonian constraint H as a condition on states in a suitable

Approaches to quantum gravity 17

Hilbert space H, that is, formally

H = 0

This is the famous WheelerDeWitt equation or quantum Einsteinequation of canonical quantum gravity and resembles a Schrodingerequation, only that the familiar /t term is missing, one of severaloccurrences of the absence or problem of time in this approach (see,e.g., [86] and references therein).

Since the status of this programme, that is, its Loop Quantum Grav-ity (LQG) incarnation, is the subject of the present book we will notgo too much into details here. The successes of LQG are a mathemati-cally rigorous framework, manifest background independence, a manifestlynon-perturbative language, an inherent notion of quantum discreteness ofspacetime which is derived rather than postulated, certain UV nitenessresults, a promising path integral formulation (spin foams) and nally aconsistent formulation of quantum black hole physics. A conceptually verysimilar but technically dierent canonical programme has been launchedby Klauder [8791] to which the following remarks apply as well.

The following issues are at the moment unresolved within this approach:* Tremendously non-linear structure

The WheelerDeWitt operator is, in the so-called ADM formulation,a functional dierential operator of second order of the worst kind,namely with non-polynomial, not even analytic (in the basic congura-tion variables) coecients. To even dene such an operator rigorouslyhas been a major problem for more than 60 years. What should be asuitable Hilbert space that carries such an operator? It is known thata