Constraint algebra in LQG reloaded : Toy model of a U(1)^{3} Gauge Theory I

Abstract

We analyze the issue of anomaly-free representations of the constraint algebra in Loop Quantum Gravity (LQG) in the context of a diffeomorphism-invariant gauge theory in three spacetime dimensions. We construct a Hamiltonian constraint operator whose commutator matches with a quantization of the classical Poisson bracket involving structure functions. Our quantization scheme is based on a geometric interpretation of the Hamiltonian constraint as a generator of phase space-dependent diffeomorphisms. The resulting Hamiltonian constraint at finite triangulation has a conceptual similarity with the "mu-bar"-scheme in loop quantum cosmology and highly intricate action on the spin-network states of the theory. We construct a subspace of non-normalizable states (distributions) on which the continuum Hamiltonian constraint is defined which leads to an anomaly-free representation of the Poisson bracket of two Hamiltonian constraints in loop quantized framework.

60 pages, 6 figures

References in corpus (6)

• Quantum Nature of the Big Bang: Improved dynamics
• Gravity quantized
• Deformed General Relativity and Effective Actions from Loop Quantum Gravity
• Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity
• Constraint algebra in LQG reloaded : Toy model of an Abelian gauge theory - II Spatial Diffeomorphisms
• Lessons for Loop Quantum Gravity from Parametrised Field Theory

Cited by in corpus (46)

• Quantum cosmology: a review
• Black Holes in Loop Quantum Gravity
• Covariance in models of loop quantum gravity: Spherical symmetry
• Anomaly-free cosmological perturbations in effective canonical quantum gravity
• Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity
• Some implications of signature-change in cosmological models of loop quantum gravity
• Time evolution as refining, coarse graining and entangling
• Hypersurface-deformation algebroids and effective space-time models
• Deformed General Relativity
• Hamiltonian operator for loop quantum gravity coupled to a scalar field
• Inhomogeneities, loop quantum gravity corrections, constraint algebra and general covariance
• Discreteness corrections and higher spatial derivatives in effective canonical quantum gravity
• Constraint algebra in LQG reloaded : Toy model of an Abelian gauge theory - II Spatial Diffeomorphisms
• Loop Quantum Gravity
• Information loss, made worse by quantum gravity?
• Group field theory and its cosmology in a matter reference frame
• Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models
• An elementary introduction to loop quantum gravity
• Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity: Diffeomorphism Covariance
• Gravitational dynamics: A novel shift in the Hamiltonian paradigm
• Minisuperspace models as infrared contributions
• The constraint algebra in Smolins' $G\to 0$ limit of 4d Euclidean Gravity
• Ising Spin Network States for Loop Quantum Gravity: a Toy Model for Phase Transitions
• Non-linear (loop) quantum cosmology
• Anomaly freedom in perturbative models of Euclidean loop quantum gravity
• Constraining the loop quantum gravity parameter space from phenomenology
• Effective constraint algebras with structure functions
• A no-singularity scenario in loop quantum gravity
• Effective Field Theory of Loop Quantum Cosmology
• Fluctuation energies in quantum cosmology
• Deriving loop quantum cosmology dynamics from diffeomorphism invariance
• Reduced Phase Space Approach to the $U(1{)}^3$ model for Euclidean Quantum Gravity
• Propagation in Polymer Parameterised Field Theory
• Deformed covariance in spherically symmetric vacuum models of loop quantum gravity: Consistency in Euclidean and self-dual gravity
• Electric Time in Quantum Cosmology
• Extending general covariance: Moyal-type noncommutative manifolds
• Inequivalence of mimetic gravity with models of loop quantum gravity
• Minisuperspace models of discrete systems
• Space-time physics in background-independent theories of quantum gravity
• Hamiltonian constraint in Euclidean LQG revisited: First hints of off-shell Closure
• A quantum kinematics for asymptotically flat spacetimes
• Exploring the Tomlin-Varadarajan quantum constraints in $U(1{)}^3$ loop quantum gravity: solutions and the Minkowski theorem
• Geometrical Quantum Time in the $U(1{)}^3$ Model of Euclidean Quantum Gravity
• On anomaly freedom in spherically symmetric lattice loop quantization
• Hypersurface deformations
• Asymptotically flat boundary conditions for the $U(1{)}^3$ model for Euclidean Quantum Gravity