The Phoenix Project: Master Constraint Programme for Loop Quantum Gravity

Abstract

The Hamiltonian constraint remains the major unsolved problem in Loop Quantum Gravity (LQG). Seven years ago a mathematically consistent candidate Hamiltonian constraint has been proposed but there are still several unsettled questions which concern the algebra of commutators among smeared Hamiltonian constraints which must be faced in order to make progress. In this paper we propose a solution to this set of problems based on the so-called {\bf Master Constraint} which combines the smeared Hamiltonian constraints for all smearing functions into a single constraint. If certain mathematical conditions, which still have to be proved, hold, then not only the problems with the commutator algebra could disappear, also chances are good that one can control the solution space and the (quantum) Dirac observables of LQG. Even a decision on whether the theory has the correct classical limit and a connection with the path integral (or spin foam) formulation could be in reach. While these are exciting possibilities, we should warn the reader from the outset that, since the proposal is, to the best of our knowledge, completely new and has been barely tested in solvable models, there might be caveats which we are presently unaware of and render the whole {\bf Master Constraint Programme} obsolete. Thus, this paper should really be viewed as a proposal only, rather than a presentation of hard results, which however we intend to supply in future submissions.

LATEX, uses AMSTEX

References in corpus (16)

• Spin Foam Models for Quantum Gravity
• Complexifier Coherent States for Quantum General Relativity
• Simplification of the Spectral Analysis of the Volume Operator in Loop Quantum Gravity
• Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework
• Spectral Analysis of the Volume Operator in Loop Quantum Gravity
• Quantum Spin Dynamics VIII. The Master Constraint
• Asymptotics of 10j symbols
• Testing the Master Constraint Programme for Loop Quantum Gravity II. Finite Dimensional Systems
• Diffeomorphism covariant representations of the holonomy-flux star-algebra
• Testing the Master Constraint Programme for Loop Quantum Gravity IV. Free Field Theories
• On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories
• Testing the Master Constraint Programme for Loop Quantum Gravity V. Interacting Field Theories
• Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models
• Renormalization of Discrete Models without Background
• Discrete quantum gravity: applications to cosmology
• Consistent discrete gravity solution of the problem of time: a model

Cited by in corpus (148)

• Background Independent Quantum Gravity: A Status Report
• Quantum Nature of the Big Bang: An Analytical and Numerical Investigation
• LQG vertex with finite Immirzi parameter
• Partial and Complete Observables for Hamiltonian Constrained Systems
• Partial and Complete Observables for Canonical General Relativity
• Fundamental Structure of Loop Quantum Gravity
• Three dimensional loop quantum gravity: physical scalar product and spin foam models
• Algebraic Quantum Gravity (AQG) I. Conceptual Setup
• Anomaly freedom in perturbative loop quantum gravity
• Spherically Symmetric Quantum Geometry: States and Basic Operators
• Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity
• Scalar Material Reference Systems and Loop Quantum Gravity
• Loop Quantum Gravity: An Inside View
• Improved and Perfect Actions in Discrete Gravity
• An effective approach to the problem of time
• Effective approach to the problem of time: general features and examples
• Simplification of the Spectral Analysis of the Volume Operator in Loop Quantum Gravity
• Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework
• Spectral Analysis of the Volume Operator in Loop Quantum Gravity
• On the regularization ambiguities in loop quantum gravity
• Loop quantization of spherically symmetric midi-superspaces
• Coarse graining methods for spin net and spin foam models
• Quantum Spin Dynamics VIII. The Master Constraint
• (Broken) Gauge Symmetries and Constraints in Regge Calculus
• Quantum-Reduced Loop Gravity: Cosmology
• Perfect discretization of reparametrization invariant path integrals
• Stepping out of Homogeneity in Loop Quantum Cosmology
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Algebraic Quantum Gravity (AQG) II. Semiclassical Analysis
• Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity I
• Extension of loop quantum gravity to $f(R)$ theories
• Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity II
• On (Cosmological) Singularity Avoidance in Loop Quantum Gravity
• Testing the Master Constraint Programme for Loop Quantum Gravity II. Finite Dimensional Systems
• New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory
• Gauge-invariant coherent states for Loop Quantum Gravity I: Abelian gauge groups
• Time evolution as refining, coarse graining and entangling
• Algebraic Quantum Gravity (AQG) III. Semiclassical Perturbation Theory
• Coarse graining free theories with gauge symmetries: the linearized case
• Testing the Master Constraint Programme for Loop Quantum Gravity IV. Free Field Theories
• Nonperturbative Loop Quantization of Scalar-Tensor Theories of Gravity
• Loop quantum $f(R)$ theories
• Quantum Cosmology: Effective Theory
• Gauge-invariant coherent states for Loop Quantum Gravity II: Non-abelian gauge groups
• Hamiltonian operator for loop quantum gravity coupled to a scalar field
• On the regularization of the constraints algebra of Quantum Gravity in 2+1 dimensions with non-vanishing cosmological constant
• Uniform discretizations: a new approach for the quantization of totally constrained systems
• Quantum Reduced Loop Gravity: Semiclassical limit
• Testing the Master Constraint Programme for Loop Quantum Gravity V. Interacting Field Theories
• Spacetime-noncommutativity regime of Loop Quantum Gravity
• The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space
• Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity
• Quantum Gravity: A Brief History of Ideas and Some Prospects
• Lattice loop quantum cosmology: scalar perturbations
• On the Physical Hilbert Space of Loop Quantum Cosmology
• Master Constraint Operator in Loop Quantum Gravity
• Critical Overview of Loops and Foams
• A regularization of the hamiltonian constraint compatible with the spinfoam dynamics
• Loop Quantum Gravity
• Hamiltonian Renormalisation I: Derivation from Osterwalder-Schrader Reconstruction
• Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy
• Linking covariant and canonical LQG II: Spin foam projector
• Towards Loop Quantum Supergravity (LQSG)
• From Classical To Quantum Gravity: Introduction to Loop Quantum Gravity
• The Fock Space of Loopy Spin Networks for Quantum Gravity
• The Holst Spin Foam Model via Cubulations
• An elementary introduction to loop quantum gravity
• Symmetry and Evolution in Quantum Gravity
• Hamiltonian formulation of unimodular gravity in the teleparallel geometry
• Semiclassical analysis of the Loop Quantum Gravity volume operator: I. Flux Coherent States
• Twistorial phase space for complex Ashtekar variables
• Quantum field theory and its symmetry reduction
• Coherent states, constraint classes, and area operators in the new spin-foam models
• On the Relation between Operator Constraint --, Master Constraint --, Reduced Phase Space --, and Path Integral Quantisation
• Matrix Elements of Lorentzian Hamiltonian Constraint in LQG
• Bouncing evolution in a model of loop quantum gravity
• 4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block
• On the geometry of quantum constrained systems
• Spin-Foam Models and the Physical Scalar Product
• The picture of the Bianchi I model via gauge fixing in Loop Quantum Gravity
• Reduced Phase Space Quantization of Black Holes: Path Integrals, and Effective Dynamics
• Discretisations, Constraints and Diffeomorphisms in Quantum Gravity
• On the structure of the space of generalized connections
• Dynamics of Scalar Field in Polymer-like Representation
• Towards the self-adjointness of a Hamiltonian operator in loop quantum gravity
• The Plebanski sectors of the EPRL vertex
• Power spectrum of primordial perturbations for an emergent universe in quantum reduced loop gravity
• Hamiltonian analysis of SO(4,1) constrained BF theory
• Euclidean LQG Dynamics: An Electric Shift in Perspective
• The Hamiltonian of Einstein affine-metric formulation of General Relativity
• On the Relation between Rigging Inner Product and Master Constraint Direct Integral Decomposition
• Rainbow-like Black Hole metric from Loop Quantum Gravity
• Towards computational insights into the large-scale structure of spin foams
• Frozen formalism and canonical quantization in group field theory
• Anisotropic loop quantum cosmology with self-dual variables
• Scalar curvature operator for models of loop quantum gravity on a cubical graph
• Loop Quantum Gravity in Ashtekar and Lagrange-Finsler Variables and Fedosov Quantization of General Relativity
• Extension of Loop Quantum Gravity to Metric Theories beyond General Relativity
• Emergent diffeomorphism invariance in a discrete loop quantum gravity model
• Automorphisms in Loop Quantum Gravity
• From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier
• Alternative dynamics in loop quantum Brans-Dicke cosmology
• Reduced Phase Space Approach to the $U(1{)}^3$ model for Euclidean Quantum Gravity
• Lessons for Loop Quantum Gravity from Parametrised Field Theory
• Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
• The closure constraint for the hyperbolic tetrahedron as a Bianchi identity
• Lessons from Toy-Models for the Dynamics of Loop Quantum Gravity
• Quantum realizations of Hilbert-Palatini second-class constraints
• Spinfoams: Foundations
• Covariant Origin of the $U(1{)}^3$ model for Euclidean Quantum Gravity
• Operators of quantum-reduced loop gravity from the perspective of full loop quantum gravity
• Loop quantum modified gravity and its cosmological application
• A Short and Subjective Introduction to the Spinfoam Framework for Quantum Gravity
• The status of Quantum Geometry in the dynamical sector of Loop Quantum Cosmology
• Uniform discretizations: a quantization procedure for totally constrained systems including gravity
• 2+1 gravity with positive cosmological constant in LQG: a proposal for the physical state
• Loop-Quantum-Gravity Simplicity Constraint as Surface Defect in Complex Chern-Simons Theory
• Intertwiner Entanglement Excitation and Holonomy Operator
• A Path-integral for the Master Constraint of Loop Quantum Gravity
• Fermion coupling to loop quantum gravity: canonical formulation
• Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit
• Dynamics in canonical models of loop quantum gravity
• Regarding the `Hole Argument' and the `Problem of Time'
• Deformation quantization of constrained systems: a group averaging approach
• A discrete, unitary, causal theory of quantum gravity
• Approximating the physical inner product of Loop Quantum Cosmology
• New formulation of Horava-Lifshitz quantum gravity as a master constraint theory
• Effective relational dynamics
• A review on Loop Quantum Gravity
• Thermodynamics in new model of loop quantum cosmology
• Taming Thiemann's Hamiltonian constraint in canonical loop quantum gravity: reversibility, eigenstates and graph-change analysis
• Spin networks on adiabatic quantum computer
• Regularized Hamiltonians and Spinfoams
• Master constraint approach to quantum-reduced loop gravity
• Quantum Cauchy Surfaces in Canonical Quantum Gravity
• Revisiting the quantum scalar field in spherically symmetric quantum gravity
• Purely geometric path integral for spin foams
• Commuting Simplicity and Closure Constraints for 4D Spin Foam Models
• Hawking radiation and Quasinormal modes
• Loop quantum gravity - a short review
• Intrinsic Time Deparameterization of The Canonical Connection Dynamics of General Relativity
• AQV V. The localised holonomy-flux cross-product *-algebra
• Covariant Quantum Gravity I: Covariant Hamiltonian Framework
• AQV VI. A holonomy groupoid formulation
• The relation between Geometry and Matter in classical and quantum Gravity and Cosmology
• The SL(2,R) totally constrained model: three quantization approaches
• Hamiltonian Theory: generalizations to higher dimensions, supersymmetry and modified gravity
• Quantum Dyanmics of Loop Quantum Gravity