Flux formulation of loop quantum gravity: Classical framework

Abstract

We recently introduced a new representation for loop quantum gravity, which is based on the BF vacuum and is in this sense much nearer to the spirit of spin foam dynamics. In the present paper we lay out the classical framework underlying this new formulation. The central objects in our construction are the so-called integrated fluxes, which are defined as the integral of the electric field variable over surfaces of codimension one, and related in turn to Wilson surface operators. These integrated flux observables will play an important role in the coarse graining of states in loop quantum gravity, and can be used to encode in this context the notion of curvature-induced torsion. We furthermore define a continuum phase space as the modified projective limit of a family of discrete phase spaces based on triangulations. This continuum phase space yields a continuum (holonomy-flux) algebra of observables. We show that the corresponding Poisson algebra is closed by computing the Poisson brackets between the integrated fluxes, which have the novel property of being allowed to intersect each other.

60 pages, 13 figures, published version

References in corpus (62)

• Background Independent Quantum Gravity: A Status Report
• Quantum Spin Dynamics (QSD)
• The Spin Foam Approach to Quantum Gravity
• Projective Techniques and Functional Integration
• Representations of the holonomy algebras of gravity and non-Abelian gauge theories
• Differential Geometry on the Space of Connections via Graphs and Projective Limits
• Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
• Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity
• Twisted geometries: A geometric parametrisation of SU(2) phase space
• Polyhedra in loop quantum gravity
• Representations of the Weyl Algebra in Quantum Geometry
• Three dimensional loop quantum gravity: physical scalar product and spin foam models
• Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles
• Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity
• Diffeomorphisms and spin foam models
• Generalized Measures in Gauge Theory
• Phase space descriptions for simplicial 4d geometries
• Improved and Perfect Actions in Discrete Gravity
• Quantum Theory of Geometry III: Non-commutativity of Riemannian Structures
• Area-angle variables for general relativity
• Quantum Spin Dynamics (QSD) : VII. Symplectic Structures and Continuum Lattice Formulations of Gauge Field Theories
• Quantum deformation of quantum gravity
• Non-commutative flux representation for loop quantum gravity
• Simplicity in simplicial phase space
• Spin Foams and Canonical Quantization
• Black Hole Entropy from complex Ashtekar variables
• Canonical quantization of general relativity in discrete space-times
• Holonomy Spin Foam Models: Definition and Coarse Graining
• Canonical Quantization of Gravitating Point Particles in 2+1 Dimensions
• Testing the Master Constraint Programme for Loop Quantum Gravity II. Finite Dimensional Systems
• Time evolution as refining, coarse graining and entangling
• Spin Networks for Non-Compact Groups
• Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint
• Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity
• Coarse graining of spin net models: dynamics of intertwiners
• Wilson surfaces and higher dimensional knot invariants
• Three dimensional loop quantum gravity: coupling to point particles
• Deformation Operators of Spin Networks and Coarse-Graining
• Loop Quantum Gravity Vacuum with Nondegenerate Geometry
• A regularization of the hamiltonian constraint compatible with the spinfoam dynamics
• CFT/Gravity Correspondence on the Isolated Horizon
• Constraint analysis for variational discrete systems
• Towards the Turaev-Viro amplitudes from a Hamiltonian constraint
• Turaev-Viro amplitudes from 2+1 Loop Quantum Gravity
• The Diffeomorphism Constraint Operator in Loop Quantum Gravity
• Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
• Loop Quantum Gravity a la Aharonov-Bohm
• Black hole horizons from within loop quantum gravity
• Dirac-like approach for consistent discretizations of classical constrained theories
• On loop quantum gravity kinematics with non-degenerate spatial background
• Three Dimensional Loop Quantum Gravity: Particles and the Quantum Double
• Chern--Simons states in spin-network quantum gravity
• Construction of spaces of kinematic quantum states for field theories via projective techniques
• Topological Lattice Gravity Using Self-Dual Variables
• Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces
• Quantization of diffeomorphism invariant theories of connections with a non-compact structure group - an example
• Consistent discretization and canonical classical and quantum Regge calculus
• Some surprising implications of background independence in canonical quantum gravity
• Near-Horizon Radiation and Self-Dual Loop Quantum Gravity
• The generator of spatial diffeomorphisms in the Koslowski- Sahlmann representation
• Some results concerning the representation theory of the algebra underlying loop quantum gravity
• Point particles in 2+1 dimensions: toward a semiclassical loop gravity formulation

Cited by in corpus (26)

• Edge modes of gravity -- II: Corner metric and Lorentz charges
• Fusion basis for lattice gauge theory and loop quantum gravity
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Effective Spin Foam Models for Four-Dimensional Quantum Gravity
• Complex network view of evolving manifolds
• On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity
• Discrete gravity dynamics from effective spin foams
• (3+1)-dimensional topological phases and self-dual quantum geometries encoded on Heegard surfaces
• Bubble Networks: Framed Discrete Geometry for Quantum Gravity
• Loop quantum gravity coupled to a scalar field
• Projective Limits of State Spaces II. Quantum Formalism
• 2+1D Loop Quantum Gravity on the Edge
• Discretization of 3d gravity in different polarizations
• Conformal boundary conditions, loop gravity and the continuum
• Quantum geometry from higher gauge theory
• From 3D topological quantum field theories to 4D models with defects
• Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases
• Corner symmetry and quantum geometry
• On a self-dual phase space for 3+1 lattice Yang-Mills theory
• The continuum approach to the BF vacuum: the U(1) case
• Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models
• Dual 2+1D Loop Quantum Gravity on the Edge
• A new realization of quantum geometry
• Loop-Quantum-Gravity Simplicity Constraint as Surface Defect in Complex Chern-Simons Theory
• Discretization of 4d Poincaré BF theory: from groups to 2-groups
• Abelian 2+1D Loop Quantum Gravity Coupled to a Scalar Field