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