0521842638 cambridge university press modern canonical quantum general relativity oct 2007

The absence of a viable quantum gravity theory to date isdue to the fact that quantum field theory as currently formulated assumes that a background geometry is available, thus being inco

GENERAL 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-stein’s 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 are

background 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 (field) theory as currently formulated assumes that

a background geometry is available, thus being inconsistent with the principles ofgeneral 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 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, so

gen-it is sugen-itable for graduate students and researchers wgen-ith a basic knowledge ofquantum 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 f¨urGravitationsphysik (Albert Einstein Institut), Potsdam, Germany He is also

a 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-Westf¨alisch Tech-nische Hochschule, Aachen, Germany He held two-year postdoctoral positions atThe Pennsylvania State University and Harvard University As of 2005 he holds

a guest professor position at Beijing Normal University, China

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 2 + 1 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 G¨ ockeler and T Sch¨ucker Differential 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 M¨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

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

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2007

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tute), 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 respec-

tively The colours or areas are quantised in units of the Planck area 2

P ≈ 10 −66cm2 Thus the faces do 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 τ P ≈ 10 −43s 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 hilated all the time Therefore, the free space not occupied by tetrahedra does not correspond

anni-to empty (matter-free) space but rather anni-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.

Foreword, by Chris Isham pagexvii

Introduction: Defining quantum gravity 1

Why quantum gravity in the twenty-first century? 1

Motivation for canonical quantum general relativity 23

I CLASSICAL FOUNDATIONS, INTERPRETATION AND THE

CANONICAL QUANTISATION PROGRAMME

1 Classical Hamiltonian formulation of General Relativity 39

1.2 Legendre transform and Dirac analysis of constraints 46

1.3 Geometrical interpretation of the gauge transformations 50

1.4 Relation between the four-dimensional diffeomorphism group and

the transformations generated by the constraints 56

1.5 Boundary conditions, gauge transformations and symmetries 60

2 The problem of time, locality and the interpretation of

2.1 The classical problem of time: Dirac observables 75

2.2 Partial and complete observables for general constrained systems 81

2.2.4 Reduced phase space quantisation of the algebra of Dirac

observables and unitary implementation of the

2.4 Quantum problem of time: physical inner product and

3 The programme of canonical quantisation 107

4 The new canonical variables of Ashtekar for

General Relativity 118

4.2.2 Canonical transformation on the extended phase space 126

II FOUNDATIONS OF MODERN CANONICAL QUANTUM

GENERAL RELATIVITY

6 Step I: the holonomy–flux algebra P 157

6.2 Definition of P: (1) Paths, connections, holonomies and

6.2.2 A natural topology on the space of generalised connections 168

6.2.3 Gauge invariance: distributional gauge transformations 175

6.2.4 The C ∗ algebraic viewpoint and cylindrical functions 183

6.3 Definition of P: (2) surfaces, electric fields, fluxes and vector fields 191

6.4 Definition of P: (3) regularisation of the holonomy–flux

6.5 Definition of P: (4) Lie algebra of cylindrical functions and

7 Step II: quantum∗-algebra A 206

8 Step III: representation theory of A 212

8.2.1 Regular Borel measures on the projective limit:

8.2.2 Functional calculus on a projective limit 226

8.2.3 + Density and support properties ofA, A/G with respect

8.2.4 Spin-network functions and loop representation 237

8.2.5 Gauge and diffeomorphism invariance of μ0 242

8.2.6 + Ergodicity of μ0with respect to spatial diffeomorphisms 245

8.2.7 Essential self-adjointness of electric flux momentum

9 Step IV: (1) implementation and solution of the

kinematical constraints 264

9.1.1 Derivation of the Gauß constraint operator 264

9.1.2 Complete solution of the Gauß constraint 266

9.2 Implementation of the spatial diffeomorphism constraint 269

9.2.1 Derivation of the spatial diffeomorphism constraint

9.2.2 General solution of the spatial diffeomorphism constraint 271

10 Step IV: (2) implementation and solution of the

Hamiltonian constraint 279

10.2 Heuristic explanation for UV finiteness due to background

10.3 Derivation of the Hamiltonian constraint operator 286

10.4 Mathematical definition of the Hamiltonian constraint operator 291

10.5 The kernel of the Wheeler–DeWitt constraint operator 311

10.6.1 Motivation for the Master Constraint Programme in

10.6.3 Physical inner product and Dirac observables 326

10.7.2 Testing the new regularisation technique by models of

10.7.3 Quantum Poincar´e algebra 341

10.7.4 Vasiliev invariants and discrete quantum gravity 344

11 Step V: semiclassical analysis 345

11.2.1 Semiclassical states and coherent states 354

11.2.2 Construction principle: the complexifier method 356

11.2.3 Complexifier coherent states for diffeomorphism-invariant

11.2.5 Semiclassical limit of loop quantum gravity: graph-changing

operators, shadows and diffeomorphism-invariant

11.2.6 + The infinite tensor product extension 385

11.3 Graviton and photon Fock states from L2( A, dμ0) 390

III PHYSICAL APPLICATIONS

12 Extension to standard matter 399

12.1 The classical standard model coupled to gravity 400

12.2 Kinematical Hilbert spaces for diffeomorphism-invariant theories

12.2.3 Gauge and diffeomorphism-invariant subspace 417

12.3 Quantisation of matter Hamiltonian constraints 418

12.3.1 Quantisation of Einstein–Yang–Mills theory 419

13 Kinematical geometrical operators 431

13.4.2 Symmetry, positivity and self-adjointness 448

13.5 Uniqueness of the volume operator, consistency with the flux

13.6 Spatially diffeomorphism-invariant volume operator 455

14 Spin foam models 458

14.1 Heuristic motivation from the canonical framework 458

14.3.1 Plebanski action and simplicity constraints 466

14.3.3 Discretisation and quantisation of BF theory 476

14.3.5 Summary of the status of the Barrett–Crane model 494

14.4 Triangulation dependence and group field theory 495

15 Quantum black hole physics 511

15.1.2 Event horizons, trapped surfaces and apparent horizons 517

15.1.3 Trapping, dynamical, non-expanding and (weakly) isolated

15.1.4 Spherically symmetric isolated horizons 526

15.1.5 Boundary symplectic structure for SSIHs 535

15.2 Quantisation of the surface degrees of freedom 540

15.2.1 Quantum U(1) Chern–Simons theory with punctures 541

15.3 Implementing the quantum boundary condition 546

15.4.1 Remaining U(1) gauge transformations 549

15.4.2 Remaining surface diffeomorphism transformations 550

16 Applications to particle physics and quantum cosmology 562

17 Loop Quantum Gravity phenomenology 572

IV MATHEMATICAL TOOLS AND THEIR CONNECTION

TO PHYSICS

18 Tools from general topology 577

19 Differential, Riemannian, symplectic and complex

20 Semianalytic category 627

21 Elements of fibre bundle theory 634

21.1 General fibre bundles and principal fibre bundles 634

22 Holonomies on non-trivial fibre bundles 644

23 Geometric quantisation 652

24 The Dirac algorithm for field theories with constraints 671

24.2 First- and second-class constraints and the Dirac bracket 674

25 Tools from measure theory 680

26 Key results from functional analysis 689

26.8 Quadratic forms 699

27 Elementary introduction to Gel’fand theory for

Abelian C∗-algebras 701

27.2 The Gel’fand transform and the Gel’fand isomorphism 709

28 Bohr compactification of the real line 713

29 Operator ∗-algebras and spectral theorem 719

29.1 Operator∗-algebras, representations and GNS construction 719

29.2 Spectral theorem, spectral measures, projection valued measures,

30 Refined algebraic quantisation (RAQ) and direct integral

decomposition (DID) 729

31 Basics of harmonic analysis on compact Lie groups 746

32 Spin-network functions for SU(2) 755

32.1 Basics of the representation theory of SU(2) 755

32.2 Spin-network functions and recoupling theory 757

32.3 Action of holonomy operators on spin-network functions 762

33 + Functional analytic description of classical connection

33.1 Infinite-dimensional (symplectic) manifolds 770

Over half a century of collective study has not diminished the fascination ofsearching for a consistent theory of quantum gravity I first 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 field 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 field theory schemes foundered on intractable 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

ultra-However, notwithstanding my early training as a quantum field 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 Wheeler–DeWitt equation arguably one ofthe most elegant equations in theoretical physics, and certainly one of the mostmathematically ill-defined Indeed, it was the very intractability of this equationthat first intrigued me and prompted me to see what could be done with moresophisticated quantization methods After much effort it became clear that theanswer was “not much.”

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

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

prob-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 first 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 Wheeler–DeWitt equation, and

he has gone on since that time to become one of the internationally acknowledgedexperts in loop quantum gravity

Thiemann’s deep knowledge of mathematics applied to quantum gravity isevident from the first page of this magnificent book The subject is explored

in considerable generality and with real mathematical depth The author startsfrom first 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 differentialgeometry 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 define thetheory For some years now I have replied to the effect that, if anybody can do

it, 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 why

I place so much trust in the author’s ability to crack this central problem ofquantum gravity

Chris Isham,Professor of Theoretical Physics atThe Blankett Laboratory, Imperial College, London

Quantum General Relativity (QGR) or Quantum Gravity for short is, by inition, a Quantum (Field) Theory of Einstein’s 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 effort 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.

def-What we do have today are candidate theories which display some promisingfeatures that one intuitively expects from a quantum theory of gravity They are

so far candidates only because for each of them one still has to show, at the end

of 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 Yang–Mills 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 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 field has now matured enough in order to justify the publication of

canon-a new textbook The litercanon-ature on LQG now comprises more thcanon-an canon-a thouscanon-and

articles scattered over a vast number of journals, reviews, proceedings and 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 a

con-‘geodesic’ through the literature enabling the reader to move quickly from thebasics to the frontiers of current research By definition, a geodesic cannot touch

on all the subjects of the theory and we apologise herewith to our colleagues

if we were unable to cover their work in this single volume manuscript 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

How-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 unified 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 only

in four spacetime dimensions and in that sense is a prediction, higher p-form

formulations in higher dimensions are conceivable Matter and geometry arenot unified in the sense that they are components of one and the same geo-metrical object, however, they are unified under the four-dimensional diffeo-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 differ 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 but

a necessity: in a field 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 or

of 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 Heisenberg’s discovery that the

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

http://www.matmor.unam.mx/corichi/lqgbib.pdf.

Bohr–Sommerfeld 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 10−33cm seems to be a prediction of the theorywhich is a first substantial evidence for a theory in which the gravitational fieldacts as a natural cutoff 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 definitions 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 amount

of 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 effort to stress the basic principles

of canonical QGR, of which LQG is just one possible incarnation based on aspecific choice of variables

For readers who want to get acquainted first with the physical ideas and 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’)

con-of a general presentation con-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 [4–14] Thestatus of the theory a decade ago is summarised in the books [15–17]

The present text is aimed at all readers who want to find 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 material

of this book in a two-semester course to German students in physics and 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 frontier

mathe-of 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 version

of this book (http://www.arxiv.org/list/gr-qc/0110034) by countless colleagues

is gratefully acknowledged, in particular those by J¨urgen Ehlers, ChristianFleischhack, Stefan Hofmann, Chris Isham, Jurek Lewandowski, Robert Oeckl,Hendryk Pfeiffer, Carlo Rovelli, Hanno Sahlmann and Oliver Winkler Special

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 figures.

Posvwa svoe ene Tatne

Ebenso gewidmet meinen S¨ohnen Andreas und Maximilian

Thomas ThiemannBerlin, Toronto 2001–2007

μ, ν, ρ, = 0, 1, , D tensorial spacetime indices

a, b, c, = 1, , D tensorial spatial indices

 a1 a D Levi–Civita totally skew tensor pseudo density

of weight−1

q ab spatial (intrinsic) metric tensor of σ

h mn , m, n, o, = 1, , N matrix elements for general G

I, J, K, = 1, 2, , dim (G) Lie algebra indices for general G

for G

[τ I , τ J ] = 2f IJ K τ K structure constants for G

π(h) (irreducible) representations for general G or

algebra

h AB , A, B, C, = 1, 2 matrix elements for SU(2)

i, j, k, = 1, 2, 3 Lie algebra indices for SU(2)

k ij = δ ij Cartan–Killing metric for SU(2)

f ij k =  ijk structure constants for SU(2)

π j (h) (irreducible) representations for SU(2) with

spin j

A I a pull-back of A to σ by local section

ι A , o A ; A = 1, 2 øA ι A :=  AB

A ι B= 1: spinor dyad

¯ι A  , ¯ o A  ; A  = 1, 2 primed (complex coinjugate) spinor dyad

complexification of G

A i

associated to G-bundle under adjointrepresentation

∗E I

a1 ,a D −1 := k IJ  a1 ,a D E a D

J : pull-back of∗E to σ by

local section

associated to SU(2)-bundle under adjointrepresentation

SU(2)-bundle under the defining representation

ι ∈S n sgn(ι) T a ι(1) a ι(n):antisymmetrisation of indices

A space of distributional connections

gauge transformations

distributional gauge transformations

A/G space of distributional gauge equivalence classes

of connections

AC

space of distributional complex connections

equivalence classes of connections

configuration space

equivalence class of s

Γω

0 set of semianalytic, compactly supported graphs

Γω

σ set of semianalytic, countably infinite graphs

Diffω

sa(σ) group of semianalytic diffeomorphisms of σ

Diffω

sa,0 (σ) group of semianalytic diffeomorphisms of σ

connected to the identityDiffω

0(σ) group of analytic diffeomorphisms of σ

connected to the identityDiffω (σ) group of analytic diffeomorphisms of σ

algebra automorphism

γ semianalytic graph

h p (A) = A(p) holonomy of A along p

curvature two-form

L(X, Y ), L(X) linear (un)bounded operators between X, Y or

topology

H0= L2( A, dμ0) uniform measure L2space

Cyll restriction of Cyl to functions cylindrical over l

A, B abstract (∗ -)algebra or C ∗-algebra

algebra or group or characteristic function of aset

In the first 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 define what a quantum theory

of 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 andfinally in the fourth motivate our choice of canonical quantum general relativity

as opposed to other approaches

Why quantum gravity in the twenty-first century?

It is often argued that quantum gravity is not relevant for the physics of this tury because in our most powerful accelerator, the LHC to be working in 2007,

cen-we obtain energies of the order of a few 103GeV 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 particles

of energies much beyond the TeV scale; we have already observed astrophysicalparticles with energy of up to 1013GeV, 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 effects

in the close future with γ-ray burst physics and the GLAST detector Next,

quantum gravity effects in the early universe might have left their fingerprint

in 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 first 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 believed

to be due to a weakly interacting massive particle (WIMP) (possibly metric) and only about 5% is made out of baryonic matter Here ‘dark’ means

supersym-1 Recent independent observations all indicate that the expansion of the universe is currently accelerating.

that these unknown forms of matter do not radiate, they are invisible Hence we

see that we only understand 5% of the matter in the universe and at least as

far 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 at

new physics beyond the standard model and are of extreme current interest See,

for example, [18–20] 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 fields that we fully understand to date in

four dimensions are free quantum fields on four-dimensional Minkowski space.

Formulated more provocatively:

In four dimensions we only understand an (infinite) collection of uncoupled harmonic oscillators on Minkowski space!

In order to leave the domain of these rather trivial and unphysical (since interacting) quantum field theories, physicists have developed two techniques:perturbation theory and quantum field theory on curved backgrounds Thismeans the following: with respect to accelerator experiments, the most important

non-processes are scattering amplitudes between particles One can formally write

down a unitary operator that accounts for the scattering interaction betweenparticles and which maps between the well-understood free quantum field Hilbertspaces in the far past and future Famously, by Haag’s theorem [21] wheneverthat operator is really unitary, there is no interaction and if it is not unitary,then it is ill-defined giving rise to the ultraviolet divergences of ordinary QFT Infact, one can only define 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 field theory Hilbert spaces The resulting series isdivergent order by order but if the theory is ‘renormalisable’ then one can makethese orders artificially finite 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 ing) quantum fields on Minkowski spacetime for which the so-called Wightmanaxioms [21] can be verified Let us summarise them for the case of a scalar field

interact-in (D + 1)-dimensional Minteract-inkowski space:

W1 Representation

There exists a unitary and continuous representation U : P → B(H) of the

Poincar´ e groupP on a Hilbert space H.

RD+1 d D+1 xφ(x)f (x) where f ∈ S(R D+1) is a test function of rapid

decrease Then finite linear combinations of the form φ(f1) φ(f N)Ω lie

dense in H (that is, Ω is a cyclic vector) and U(p)φ(f)U(p) −1 = φ(f ◦ p)

for any p ∈ P.

W5 Locality (causality)

Suppose that the supports (the set of points where a function is

differ-ent from zero) of f, f  are spacelike separated (that is, the points of

their supports cannot be connected by a non-spacelike curve) then [φ(f ),

φ(f )] = 0

The most important objects in this list are those that are highlighted in

boldface letters: the fixed, non-dynamical Minkowski background metric η with

its well-defined causal structure, its Poincar´e symmetry groupP, the associated

representation U (p) of its elements, the invariant vacuum state Ω and finally

the fixed, non-dynamical topological, differentiable 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 Poincar´e group from which one buildsthe usual Fock Hilbert spaces of the free fields We see that the whole structure

of the theory is heavily based on the existence of these objects which come with

a fixed, non-dynamical background metric on a fixed, non-dynamical topologicaland differentiable 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 field theory [22]and the so-called microlocal analysis [23–26] (in which the locality axiom is takencare of pointwise rather than globally), which recently have also been employed

to develop perturbation theory on arbitrary background spacetimes [27–33] 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 fields, at least perturbatively, on arbitrarybackgrounds

In order to go beyond a fixed background one can consider ‘all backgroundssimultaneously’ [35, 36] Namely, the notion of a local quantum field theoryA(M, g) (thought of as a unital C∗-algebra for convenience) on a given curved

background spacetime (M, g) can be generalised in the following way:2 given

an isometric embedding ϕ : (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 α ϕ : A(M, g) → A(M  , g  ) The homomorphisms α ψ could for instancejust act geometrically by pulling back the fields 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 diffeomorphism On the other hand, we have the category Alg

whose 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 field is then a covariant functor A : Man→ Alg; (M, g) →

A(M, g), ϕ→ α ϕ which relates objects and morphisms of Man with those of

Alg The functor is called causal if those quantum field theories A(M j , g j)

for which there exist isometric embeddings ϕ j : (M j , g j)→ (M, g); j = 1, 2 so

that ϕ1(M1), ϕ2(M2) are spacelike separated with respect to g satisfy the causality axiom [α ϕ1(A(M1 , g1)), αϕ2(A(M2 , g2))] ={0} The functor is said to

obey the time slice axiom when α ϕ (A(M, g)) = A(M  , g )) for all isometries

ϕ : (M, g) → (M  , g  ) such that ϕ(M ) contains a Cauchy surface for (M  , g ).This framework is background-independent because the functor A considers all

backgrounds (M, g) simultaneously.

Unfortunately, QFT on curved spacetimes, even stated in this 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 Einstein’s equations Moreover, it neglects the fact thatthe gravitational field 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 field in the same fashion, it is also unclear whether the procedure leads

background-to (unitarily) equivalent results when using backgrounds which are physicallydifferent, such as two Schwarzschild spacetimes with different mass (the cor-

responding spacetimes are not isometric) More seriously, it is expected that

especially in extreme astrophysical or cosmological situations (black holes, big

bang) the notion of a classical, smooth spacetime breaks down altogether!

In other words, the fluctuations 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 first reading, however, the appearing notions are all explained in this book (see, e.g., Definition 6.2.6 and Chapter 29).

energy spectrum of the hydrogen atom far away from the continuum limit It isprecisely here where a full-fledged quantum theory of gravity is needed: we must

be 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 of

GR where the lightcones themselves start fluctuating and hence locality becomes

a fuzzy notion Let us phrase this again, provocatively, as:

The whole framework of ordinary quantum field theory breaks down once we make the gravitational field (and the differentiable 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 long

as the backreaction of matter on geometry can be neglected and that the 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,

cou-it seems fully sufficient to have only a classical theory of general relativcou-ity andperturbative quantum field theory on curved spacetimes

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

unsatisfac-(i) Classical geometry – quantum matter inconsistency

There 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 cannot

be neglected Namely, geometry couples to matter through Einstein’s

equations

2R · g μν = κ T μν [g]

and since matter underlies the rules of quantum mechanics, the

right-hand side of this equation, the stress–energy tensor T μν [g], becomes

an operator One has tried to keep geometry classical while matter is

quantum mechanical by replacing T μν [g] by the Minkowski vacuum Ω η

expectation value < Ω η , ˆ T μν [η]Ω η >, but the solution of this equation

will give g = η which one then has to feed back into the definition

of the vacuum expectation value, and so on Notice that the notion

of vacuum itself depends on the background metric, so that this is ahighly non-trivial iteration process The resulting iteration does not

converge in general [37] Thus, such a procedure is also inconsistent,

whence we must quantise the gravitational field as well This leads to the quantum Einstein equations

in higher loop diagrams with arbitrarily large energy Suppose that

such a particle has energy E and momentum P ≈ E/c in some rest

frame According to quantum mechanics, such a particle has a lifetime

τ ≈ ¯h/E and a spatial extension given by the Compton radius λ ≈

¯hc/E According to classical GR, such a lump of energy collapses to a

black hole if the Compton radius drops below the Schwarzschild radius

r ≈ GE/c4, in other words, when the energy exceeds the Planck energy

E p=

¯hc/Gc2 The problem is now not only that in ordinary QFT thisgeneral relativistic effect is neglected, but moreover that this effect leads

to new processes: according to the Hawking effect, after the lifetime τ

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 spacetime

distances below the Planck scale p=√

¯hcG/c2 It is due to considerations

of this kind that one expects that gravity provides a natural UV cutoff 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 see

a particular incarnation of this idea in LQG

(ii) Inherent classical geometry inconsistency

Even without quantum theory at all Einstein’s field equations predict 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 finite

space-time lapse due to emission of Bremsstrahlung) but is easily explained byquantum theory which bounds the electron’s energy from below.

(iii) Inherent quantum matter inconsistency

As outlined above, perturbative quantum field theory on curved spacetimes

is itself also ill-defined 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 infinite vacuum energies being usuallyneglected in such a procedure contribute to the cosmological constant andshould have a large gravitational backreaction effect That such energy sub-tractions are quite significant is maybe best demonstrated by the Casimireffect Now, since general relativity possesses a fundamental length scale,

the Planck length p ≈ 10 −33 cm, it has been argued ever since that itation plus matter should give a finite quantum theory since gravitationprovides the necessary, built-in, short distance cutoff

grav-(iv) Cosmological constant problem

However, that cutoff cannot work naively: consider for simplicity a free less scalar field on Minkowski space The difference between the Hamiltonianand its normal ordered version is given by the divergent expression

where Δ is the flat space Laplacian If we assume a naive momentum

cut-off due to quantum gravity at|k| ≤ 1/ P the divergent momentum integral

becomes proportional to −4 P Comparing this with the cosmological stant Hamiltonian GΛ

con-d3x

det(q) where Λ is the cosmological constant, G

is Newton’s constant and q is the spatial metric (which is flat on Minkowski space) then we conclude that Λ 2

(v) Perturbative quantum gravity inconsistency

Given 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 field theory

on curved spacetime work as well for the gravitational field Roughly, the

procedure is to write the dynamical metric tensor as g = η + h where η is the Minkowski metric and h is the deviation of g from it (the graviton) and then to expand the Lagrangian as an infinite power series in h One arrives

at a formal, infinite series with finite radius of convergence which becomesmeaningless if the fluctuations are large Although the naive power countingargument implies that general relativity so defined is a non-renormalisable

theory, it was hoped that due to cancellations of divergences the tion theory could actually be finite However, that this hope was unjustifiedwas 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.

perturba-It is well known that the (locally) supersymmetric extension of a givennon-supersymmetric field 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 finite 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 [42–44]

Summarising, although a definite 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 field theory approaches to quantum gravity are meaningless.The upshot of these considerations is that our understanding of quantum fieldtheory and therefore fundamental physics is quite limited unless one quantisesthe gravitational field as well Being very sharply critical one could say:

The current situation in fundamental physics can be compared with the one at the end of the nineteenth century: while one had a success- ful theory of electromagnetism, one could not explain the stability of atoms One did not need to worry about this from a practical point of view since atomic length scales could not be resolved at that time but from a fundamental point of view, Maxwell’s 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 discussed above and it is similarly expected that the more complete theory of quantum gravity will radically change our view of the world That

is, considering the metric as a quantum operator will bring us beyond standard 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

Newtonian physics First, the determinism of Newton’s equations of motion 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 of

evap-physics are background-independent, mathematically expressed by the classical Einstein equations which are generally (or four-diffeomorphism) covariant As we have argued, it is this new element of background independence brought in with

Einstein’s theory of gravity which completely changes our present understanding

of quantum field theory

A satisfactory physical theory must combine both of these fundamental ciples, quantum mechanics and general relativity, in a consistent way and will becalled ‘Quantum Gravity’ However, the quantisation of the gravitational fieldhas 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, none

prin-of 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 culminated

in the so-called standard model of matter, in particular, the spectacular success

of perturbative quantum electrodynamics whose theoretical predictions could be

verified 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, completely

ignores General Relativity

2 Classical General Relativity or geometry, which is a background-independent

theory of all interactions but completely ignores quantum mechanics.What is so special about the gravitational force that it has persisted in its

quantisation for about 70 years already? As outlined in the previous section, the

answer is simply that today we only know how to do QFT on fixed background metrics The whole formalism of ordinary QFT relies heavily on this background

structure and collapses to nothing when it is missing It is already much moredifficult 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 fluctuating quantum field itself This underlines once more the source of

our current problem of quantising gravity: we have to learn how to do QFT on

a differential manifold (or something even more rudimentary, not even relying

on a fixed topological, differentiable manifold) rather than a spacetime.

In order to proceed, today a high-energy physicist has the choice between

the following two, extreme approaches Either the particle physicist’s, who

prefers to take over the well-established mathematical machinery from QFT

on a background at the price of dropping background independence altogether

to begin with and then tries to find the true background-independent theory

by summing the perturbation series (summing over all possible backgrounds)

Or the quantum geometer’s, who believes that background independence lies at

the 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 Einstein’s theory associated with background independence from dif-ferent ends

The particle physicist’s 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 a

Minkowski background We see that perturbation theory, by its very definition,

breaks background independence and diffeomorphism invariance at every finite order of perturbation theory Thus one can restore background independence

only by summing up the entire perturbation series, which is of course not easy

Not surprisingly, as already mentioned, since ¯hκ = 2 has negative mass sion in Planck units, applying this programme to Einstein’s theory itself results

dimen-in a mathematical disaster, a so-called non-renormalisable theory without anypredictive power In order to employ perturbation theory, it seems that one has

to go to string theory which, however, requires the introduction of new additionalstructures that Einstein’s classical theory did not know about: supersymmetry,extra dimensions and an infinite tower of new and very heavy particles next tothe graviton This is a fascinating but extremely drastic modification of generalrelativity and one must be careful not to be in conflict 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 unified 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 geometer’s language is a non-perturbative one, keeping ground independence as a guiding principle at every stage of the construction ofthe theory, resulting in mathematical structures drastically different from theones of ordinary QFT on a background metric One takes Einstein’s theoryabsolutely seriously, uses only the principles of General Relativity and quantummechanics and lets the theory build itself, driven by mathematical consistency

back-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 conflict with phenomenology but also it does notobviously explain the particle content of the world However, it tries to unify allinteractions in a different sense: all interactions must transform under a com-mon gauge group, the four-dimensional diffeomorphism group which on the otherhand is almost completely broken in perturbative approaches

Let us remark that even without specifying further details, any QGR theory

is 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 series

of these renormalised, finite terms diverges, one says the theory is not finite.The first, 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 simply

no Feynman diagrams! Thus, the mere existence of a consistent independent quantum gravity theory could imply a finite quantum theory of allinteractions Of course, a successful quantum gravity theory must recover all the

background-results that have been obtained by perturbative techniques and that have been

verified 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 No

matter 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 field 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 effective 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 Yang–Mills 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 theory

The only known consistent perturbative approach to quantum gravity is stringtheory which has good chances to be a theory that unifies all interactions.String theory [45] is not a field theory in the ordinary sense of the word

Originally, it was a two-dimensional field theory of worldsheets embedded

into a fixed, D-dimensional pseudo-Riemannian manifold (M, g) of Lorentzian

signature which is to be thought of as the spacetime of the physical world

The Lagrangian of the theory is a kind of non-linear σ-model Lagrangian for the associated embedding variables X (and their supersymmetric partners

in case of the superstring) If one perturbs g(X) = η + h(X) as above and keeps only the lowest order in X one obtains a free field 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 significant 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 interprets

as the graviton Until recently, the superstring was favoured since only therewas it believed to be possible to get rid of an unstable tachyonic vacuum state

by 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 string

action, sufficient for one-loop corrections, into the associated path integralone finds a consistent quantum theory up to one loop only if the backgroundmetric satisfies the Einstein equations These are the most powerful outcomes

of the theory: although one started out with a fixed background metric, thebackground is not arbitrary but has to satisfy the Einstein equations up tohigher loop corrections, indicating that the one-loop effective action for the

low-energy quantum field theory in those D dimensions is Einstein’s theory

plus corrections Finally, only recently has it been shown [47] that at leastthe type II superstring theories are one- and two-loop and, possibly, to all

orders, finite String theorists therefore argue to have found candidates for a

consistent theory of quantum gravity with the additional advantage that they

do 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 that one has to argue that the extra D − 3 dimensions are ‘tiny’ in the Kaluza–

Klein sense although nobody knows the mechanism responsible for this

‘spontaneous compactification’ According to [49] there exist at least 104consistent, distinct Calabi–Yau compactifications (other compactificationssuch as toroidal ones seem to be inconsistent with phenomenology), each

of which has an order of 102free, continuous parameters (moduli) like thevacuum expectation value of the Higgs field in the standard model For each

compactification of each of the five string theories in D = 10 dimensions

and for each choice of the moduli one obtains a distinct low-energy effectivetheory 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 pretation of string theory but it is conceptually simplified 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 strong–weak coupling) duality transformations between all thesestring theories exist, which suggests that we do not have 104unrelated 102-dimensional moduli spaces but that rather these 102-dimensional manifoldsintersect in singular, lower-dimensional submanifolds corresponding to cer-tain singular moduli configurations 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 field)

inter-Crucial in this picture are so-called D-branes, higher-dimensional objects

additional 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 electric–magnetic duality of Maxwell theory underwhich strong and weak coupling are exchanged Further relations betweendifferent 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 compactification 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 one

of these singular limits since the size of the 11th compactified dimension

is 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 match

Until today, no conclusive proof exists that for any of the compactificationsdescribed above we obtain a low-energy effective 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 also

an infinite 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 Kaluza–Klein modes whose masses are inverse