Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter

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
20.3 Applications
20.3.1 Classical relativity
20.3.2 The problem of time
20.3.3 Cosmological applications
20.3.4 Fundamental decoherence, black hole information puzzle, limitations to quantum computing
20.4 Constructing the quantum theory
20.5 The quantum continuum limit
20.6 Summary and outlook
References
21 The causal set approach to Quantum Gravity
21.1 The causal set approach
21.1.1 Arguments for spacetime discreteness
21.1.2 What kind of discreteness?
21.1.3 The continuum approximation
21.1.4 Reconstructing the continuum
21.1.5 Lorentz invariance and discreteness
21.1.6 LLI and discreteness in other approaches
21.2 Causal set dynamics
21.2.1 Growth models
21.2.2 Actions and amplitudes
21.3 Causal set phenomenology
21.3.1 Predicting Lambda
21.3.2 Swerving particles and almost local fields
21.4 Conclusions
References
Questions and answers
Part V Effective models and Quantum Gravity phenomenology
22 Quantum Gravity phenomenology
22.1 The “Quantum Gravity problem”, as seen by a phenomenologist
22.1.1 Quantum Gravity phenomenology exists
22.1.2 Task one accomplished: some effects introduced genuinely at the Planck scale could be seen
22.1.3 Concerning task two
22.1.4 Neutrinos and task three
22.2 Concerning Quantum Gravity effects and the status of Quantum Gravity theories
22.2.1 Planck-scale departures from classical spacetime symmetries
22.2.2 Planck-scale departures from CPT symmetry
22.2.3 Distance fuzziness and spacetime foam
22.2.4 Decoherence
22.2.5 Planck-scale departures from the equivalence principle
22.2.6 Critical-dimension superstring theory
22.2.7 Loop quantum gravity
22.2.8 Approaches based on noncommutative geometry
22.3 On the status of different areas of Quantum Gravity phenomenology
22.3.1 Planck-scale modifications of Poincaré symmetries
22.3.2 Planck-scale modifications of CPT symmetry and decoherence
22.3.3 Distance fuzziness and spacetime foam
22.3.4 Decoherence
22.3.5 Planck-scale departures from the equivalence principle
22.4 Aside on doubly special relativity: DSR as seen by the phenomenologist
22.4.1 Motivation
22.4.2 Defining the DSR scenario
22.5 More on the phenomenology of departures from Poincaré symmetry
22.5.1 On the test theories with modified dispersion relation
22.5.2 Photon stability
22.5.3 Threshold anomalies
22.5.4 Time-of-travel analyses
22.5.5 Synchrotron radiation
22.6 Closing remarks
References
23 Quantum Gravity and precision tests
23.1 Introduction
23.2 Non-renormalizability and the low-energy approximation
23.2.1 A toy model
23.2.1.1 Spectrum and scattering
23.2.1.2 The low-energy effective theory
23.2.2 Computing loops
23.2.3 The effective Lagrangian logic
23.3 Gravity as an effective theory
23.3.1 The effective action
23.3.2 Power counting
23.4 Summary
Acknowledgements
References
24 Algebraic approach to Quantum Gravity II: noncommutative spacetime
24.1 Introduction
24.2 Basic framework of NCG
24.3 Bicrossproduct quantum groups and matched pairs
24.3.1 Nonlinear factorisation in the 2D bicrossproduct model
24.3.2 Bicrossproduct Ulambda (poinc1,1) quantum group
24.3.3 Bicrossproduct Clambda[Poinc] quantum group
24.4 Noncommutative spacetime, plane waves and calculus
24.5 Physical interpretation
24.5.1 Prequantum states and quantum change of frames
24.5.2 The …-product, classicalisation and effective actions
24.6 Other noncommutative spacetime models
References
25 Doubly special relativity
25.1 Introduction: what is DSR?
25.2 Gravity as the origin of DSR
25.3 Gravity in 2+1 dimensions as DSR theory
25.4 Four dimensional field theory with curved momentum space
25.5 DSR phenomenology
25.6 DSR – facts and prospects
Acknowledgement
References
26 From quantum reference frames to deformed special relativity
26.1 Introduction
26.2 Physics of Quantum Gravity: quantum reference frame
26.3 Semiclassical spacetimes
26.3.1 Modified measurement
26.3.2 Spacetimes reconstruction
26.3.2.1 Finsler geometry
26.3.2.2 Extended phase space
26.3.3 Multiparticles states
26.4 Conclusion
Acknowledgements
References
27 Lorentz invariance violation and its role in Quantum Gravity phenomenology
27.1 Introduction
27.2 Phenomenological models
27.3 Model calculation
27.4 Effective long-distance theories
27.5 Difficulties with the phenomenological models
27.6 Direct searches
27.7 Evading the naturalness argument within QFT
27.8 Cutoffs in QFT and the physical regularization problem
27.9 Discussion
Acknowledgments
References
28 Generic predictions of quantum theories of gravity
28.1 Introduction
28.2 Assumptions of background independent theories
28.3 Well studied generic consequences
28.3.1 Discreteness of quantum geometry and ultraviolet finiteness
28.3.2 Elimination of spacetime singularities
28.3.3 Entropy of black hole and cosmological horizons
28.3.4 Heat and the cosmological constant
28.4 The problem of the emergence of classical spacetime
28.5 Possible new generic consequences
28.5.1 Deformed Special Relativity
28.5.2 Emergent matter
28.5.3 Disordered locality
28.5.4 Disordered locality and the CMB spectrum
28.6 Conclusions
Acknowledgements
References
Questions and answers
Index