Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Contributors -- Preface -- Part I Fundamental ideas and general formalisms -- 1 Unfinished revolution -- 1.1 Quantum spacetime -- 1.1.1 Space -- 1.1.2 Time -- 1.1.3 Conceptual issues -- 1.2 Where are we? -- Bibliographical note -- References -- 2 The fundamental nature of space and time -- 2.1 Quantum Gravity as a non-renormalizable gauge theory -- 2.2 A prototype: gravitating point particles in 2 + 1 dimensions -- 2.3 Black holes, causality and locality -- 2.4 The only logical way out: deterministic quantum mechanics -- 2.5 Information loss and projection -- 2.6 The vacuum state and the cosmological constant -- 2.7 Gauge- and diffeomorphism invariance as emergent symmetries -- References -- 3 Does locality fail at intermediate length scales? -- 3.1 Three D'Alembertians for two-dimensional causets -- 3.1.1 First approach through the Green function -- 3.1.2 Retarded couplings along causal links -- 3.1.3 Damping the fluctuations -- 3.2 Higher dimensions -- 3.3 Continuous nonlocality, Fourier transforms and stability -- 3.3.1 Fourier transform methods more generally -- 3.4 What next? -- 3.5 How big is lambda0? -- Acknowledgements -- References -- 4 Prolegomena to any future Quantum Gravity -- 4.1 Introduction -- 4.1.1 Background dependence versus background independence -- 4.1.2 The primacy of process -- 4.1.3 Measurability analysis -- 4.1.4 Outline of the chapter -- 4.2 Choice of variables and initial value problems in classical electromagnetic theory -- 4.3 Choice of fundamental variables in classical GR -- 4.3.1 Metric and affine connection -- 4.3.2 Projective and conformal structures -- 4.4 The problem of Quantum Gravity -- 4.5 The nature of initial value problems in General Relativity -- 4.5.1 Constraints due to invariance under a function group.

4.5.2 Non-dynamical structures and differential concomitants -- 4.6 Congruences of subspaces and initial-value problems in GR -- 4.6.1 Vector fields and three-plus-one initial value problems -- 4.6.2 Simple bivector fields and two-plus-two initial value problems -- 4.6.3 Dynamical decomposition of metric and connection -- 4.7 Background space-time symmetry groups -- 4.7.1 Non-maximal symmetry groups and partially fixed backgrounds -- 4.7.2 Small perturbations and the return of diffeomorphism invariance -- 4.7.3 Asymptotic symmetries -- 4.8 Conclusion -- Acknowledgements -- References -- 5 Spacetime symmetries in histories canonical gravity -- 5.1 Introduction -- 5.1.1 The principles of General Relativity -- 5.1.2 The histories theory programme -- 5.2 History Projection Operator theory -- 5.2.1 Consistent histories theory -- 5.2.2 HPO formalism - basics -- 5.2.3 Time evolution - the action operator -- Relativistic quantum field theory -- 5.3 General Relativity histories -- 5.3.1 Relation between spacetime and canonical description -- The representation of the group Diff(M) -- Canonical description -- 5.3.2 Invariance transformations -- Equivariance condition -- Relation between the invariance groups -- 5.3.3 Reduced state space -- 5.4 A spacetime approach to Quantum Gravity theory -- 5.4.1 Motivation -- 5.4.2 Towards a histories analogue of loop quantum gravity -- Acknowledgement -- References -- 6 Categorical geometry and the mathematical foundations of Quantum Gravity -- 6.1 Introduction -- 6.2 Some mathematical approaches to pointless space and spacetime -- 6.2.1 Categories in quantum physics Feynmanology -- 6.2.2 Grothendieck sites and topoi -- 6.2.3 Higher categories as spaces -- 6.2.4 Stacks and cosmoi -- 6.3 Physics in categorical spacetime -- 6.3.1 The BC categorical state sum model -- 6.3.2 Decoherent histories and topoi.

6.3.3 Application of decoherent histories to the BC model -- 6.3.4 Causal sites -- 6.3.5 The 2-stack of Quantum Gravity? Further directions -- Acknowledgements -- References -- 7 Emergent relativity -- 7.1 Introduction -- 7.2 Two views of time -- 7.2.1 Fermi points -- 7.2.2 Quantum computation -- 7.3 Internal Relativity -- 7.3.1 Manifold matter -- 7.3.2 Metric from dynamics -- 7.3.3 The equivalence principle and the Einstein equations -- 7.3.4 Consequences -- 7.4 Conclusion -- References -- 8 Asymptotic safety -- 8.1 Introduction -- 8.2 The general notion of asymptotic safety -- 8.3 The case of gravity -- 8.4 The Gravitational Fixed Point -- 8.5 Other approaches and applications -- 8.6 Acknowledgements -- References -- 9 New directions in background independent Quantum Gravity -- 9.1 Introduction -- 9.2 Quantum Causal Histories -- 9.2.1 Example: locally evolving networks of quantum systems -- 9.2.2 The meaning of Gamma -- 9.3 Background independence -- 9.4 QCH as a discrete Quantum Field Theory -- 9.5 Background independent theories of quantum geometry -- 9.5.1 Advantages and challenges of quantum geometry theories -- 9.6 Background independent pre-geometric systems -- 9.6.1 The geometrogenesis picture -- 9.6.2 Advantages and challenges of pre-geometric theories -- 9.6.3 Conserved quantities in a BI system -- 9.7 Summary and conclusions -- References -- Questions and answers -- Part II String/M-theory -- 10 Gauge/gravity duality -- 10.1 Introduction -- 10.2 AdS/CFT duality -- 10.3 Lessons, generalizations, and open questions -- 10.3.1 Black holes and thermal physics -- 10.3.2 Background independence and emergence -- 10.3.3 Generalizations -- 10.3.4 Open questions -- Acknowledgments -- References -- 11 String theory, holography and Quantum Gravity -- 11.1 Introduction -- 11.2 Dynamical constraints -- 11.3 Quantum theory of de Sitter space.

11.4 Summary -- References -- 12 String field theory -- 12.1 Introduction -- 12.2 Open string field theory (OSFT) -- 12.2.1 Witten's cubic OSFT action -- 12.2.2 The Sen conjectures -- 12.2.3 Outstanding problems and issues in OSFT -- 12.3 Closed string field theory -- 12.4 Outlook -- Acknowledgements -- References -- Questions and answers -- Part III Loop quantum gravity and spin foam models -- 13 Loop quantum gravity -- 13.1 Introduction -- 13.2 Canonical quantisation of constrained systems -- 13.3 Loop quantum gravity -- 13.3.1 New variables and the algebra… -- 13.3.1.1 The quantum algebra…and its representations -- 13.3.1.2 Implementation and solution of the constraints -- 13.3.2 Outstanding problems and further results -- References -- 14 Covariant loop quantum gravity? -- 14.1 Introduction -- 14.2 Lorentz covariant canonical analysis -- 14.2.1 Second class constraints and the Dirac bracket -- 14.2.2 The choice of connection and the area spectrum -- 14.3 The covariant connection and projected spin networks -- 14.3.1 A continuous area spectrum -- 14.3.2 Projected spin networks -- 14.3.3 Simple spin networks -- 14.4 Going down to SU(2) loop gravity -- 14.5 Spin foams and the Barrett-Crane model -- 14.5.1 Gravity as a constrained topological theory -- 14.5.2 Simple spin networks again -- 14.5.3 The issue of the second class constraints -- 14.6 Concluding remarks -- References -- 15 The spin foam representation of loop quantum gravity -- 15.1 Introduction -- 15.2 The path integral for generally covariant systems -- 15.3 Spin foams in 3d Quantum Gravity -- 15.3.1 The classical theory -- 15.3.2 Spin foams from the Hamiltonian formulation -- 15.3.3 The spin foam representation -- 15.3.4 Quantum spacetime as gauge-histories -- 15.4 Spin foam models in four dimensions -- Spin foam representation of canonical LQG.

Spin foam representation in the Master Constraint Program -- Spin foam representation: the covariant perspective -- 15.4.1 The UV problem in the background independent context -- Acknowledgement -- References -- 16 Three-dimensional spin foam Quantum Gravity -- 16.1 Introduction -- 16.2 Classical gravity and matter -- 16.3 The Ponzano-Regge model -- 16.3.1 Gauge symmetry -- 16.4 Coupling matter to Quantum Gravity -- 16.4.1 Mathematical structure -- 16.5 Quantum Gravity Feynman rules -- 16.5.1 QFT as the semi-classical limit of QG -- 16.5.2 Star product -- 16.6 Effective non-commutative field theory -- 16.7 Non-planar diagrams -- 16.8 Generalizations and conclusion -- Acknowledgements -- References -- 17 The group field theory approach to Quantum Gravity -- 17.1 Introduction and motivation -- 17.2 The general formalism -- 17.3 Some group field theory models -- 17.4 Connections with other approaches -- 17.5 Outlook -- References -- Questions and answers -- Part IV Discrete Quantum Gravity -- 18 Quantum Gravity: the art of building spacetime -- 18.1 Introduction -- 18.2 Defining CDT -- 18.3 Numerical analysis of the model -- 18.3.1 The global dimension of spacetime -- 18.3.2 The effective action -- 18.3.3 Minisuperspace -- 18.4 Discussion -- Acknowledgments -- References -- 19 Quantum Regge calculus -- 19.1 Introduction -- 19.2 The earliest quantum Regge calculus: the Ponzano-Regge model -- 19.3 Quantum Regge calculus in four dimensions: analytic calculations -- 19.4 Regge calculus in quantum cosmology -- 19.5 Matter fields in Regge calculus and the measure -- 19.6 Numerical simulations of discrete gravity using Regge calculus -- 19.7 Canonical quantum Regge calculus -- 19.8 Conclusions -- Acknowledgements -- References -- 20 Consistent discretizations as a road to Quantum Gravity -- 20.1 Consistent discretizations: the basic idea.

20.2 Consistent discretizations.