A symmetric scalar constraint for loop quantum gravity

Abstract

In the framework of loop quantum gravity, we define a new Hilbert space of states which are solutions of a large number of components of the diffeomorphism constraint. On this Hilbert space, using the methods of Thiemann, we obtain a family of gravitational scalar constraints. They preserve the Hilbert space for every choice of lapse function. Thus adjointness and commutator properties of the constraint can be investigated in a straightforward manner. We show how the space of solutions of the symmetrized constraint can be defined by spectral decomposition, and the Hilbert space of physical states by subsequently fully implementing the diffeomorphism constraint. The relationship of the solutions to those resulting from a proposal for a symmetric constraint operator by Thiemann remains to be elucidated.

10 pages, v2: added some references and explanations. This version is identical to the published one up to cosmetics

References in corpus (17)

• Background Independent Quantum Gravity: A Status Report
• Quantum Spin Dynamics (QSD)
• Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
• Differential Geometry on the Space of Connections via Graphs and Projective Limits
• Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
• Fundamental Structure of Loop Quantum Gravity
• Quantum Spin Dynamics (QSD) II
• The physical hamiltonian in nonperturbative quantum gravity
• QSD III : Quantum Constraint Algebra and Physical Scalar Product in Quantum General Relativity
• Quantum Spin Dynamics (QSD) : VII. Symplectic Structures and Continuum Lattice Formulations of Gauge Field Theories
• On the consistency of the constraint algebra in spin network quantum gravity
• Loop constraints: A habitat and their algebra
• Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity
• A curvature operator for LQG
• The Hamiltonian constraint in Polymer Parametrized Field Theory
• The Diffeomorphism Constraint Operator in Loop Quantum Gravity
• Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity: Diffeomorphism Covariance

Cited by in corpus (28)

• New scalar constraint operator for loop quantum gravity
• Hamiltonian operator for loop quantum gravity coupled to a scalar field
• New Hamiltonian constraint operator for loop quantum gravity
• An elementary introduction to loop quantum gravity
• First-Order Quantum Correction in Coherent State Expectation Value of Loop-Quantum-Gravity Hamiltonian
• Loop quantum gravity coupled to a scalar field
• Towards the self-adjointness of a Hamiltonian operator in loop quantum gravity
• Euclidean LQG Dynamics: An Electric Shift in Perspective
• Spin-foam model for gravity coupled to massless scalar field
• Propagation in Polymer Parameterised Field Theory
• On quantum propagation in Smolin's weak coupling limit of 4d Euclidean Gravity
• From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier
• On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity
• On the polymer quantization of connection theories: graph coherent states
• Graph coherent states for 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
• Axisymmetric gravity in real Ashtekar variables: the quantum theory
• Dynamics in canonical models of loop quantum gravity
• Scalar curvature operator for quantum-reduced loop gravity
• Exploring the Tomlin-Varadarajan quantum constraints in $U(1{)}^3$ loop quantum gravity: solutions and the Minkowski theorem
• Homogeneous-isotropic sector of loop quantum gravity: new approach
• Loop Quantum Vector-Tensor Gravity and Its Spherically Symmetric Model
• A Fock space structure for the diffeomorphism invariant Hilbert space of loop quantum gravity and its applications
• Properties of the Rovelli-Smolin-DePietri volume operator in the spaces of monochromatic intertwiners
• Deparametrization and quantization of scalar-tensor gravity and its cosmological model
• Realization of Dirac quantization in loop quantum gravity
• On the dynamics of single-vertex states in quantum-reduced loop gravity