A quantum reduction to spherical symmetry in loop quantum gravity

Abstract

Based on a recent purely geometric construction of observables for the spatial diffeomorphism constraint, we propose two distinct quantum reductions to spherical symmetry within full 3+1-dimensional loop quantum gravity. The construction of observables corresponds to using the radial gauge for the spatial metric and allows to identify rotations around a central observer as unitary transformations in the quantum theory. Group averaging over these rotations yields our first proposal for spherical symmetry. Hamiltonians of the full theory with angle-independent lapse preserve this spherically symmetric subsector of the full Hilbert space. A second proposal consists in implementing the vanishing of a certain vector field in spherical symmetry as a constraint on the full Hilbert space, leading to a close analogue of diffeomorphisms invariant states. While this second set of spherically symmetric states does not allow for using the full Hamiltonian, it is naturally suited to implement the spherically symmetric midisuperspace Hamiltonian, as an operator in the full theory, on it. Due to the canonical structure of the reduced variables, the holonomy-flux algebra behaves effectively as a one parameter family of 2+1-dimensional algebras along the radial coordinate, leading to a diagonal non-vanishing volume operator on 3-valent vertices. The quantum dynamics thus becomes tractable, including scenarios like spherically symmetric dust collapse.

5 pages

References in corpus (23)

• Quantum Spin Dynamics (QSD)
• Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
• Dust as a Standard of Space and Time in Canonical Quantum Gravity
• Geometrodynamics of Schwarzschild Black Holes
• Differential Geometry on the Space of Connections via Graphs and Projective Limits
• Loop quantization of the Schwarzschild black hole
• A Quantum Gravity Extension of the Inflationary Scenario
• Actions for Gravity, with Generalizations: A Review
• Time and a physical Hamiltonian for quantum gravity
• Canonical Quantization of Spherically Symmetric Gravity in Ashtekar's Self-Dual Representation
• Black hole entropy and SU(2) Chern-Simons theory
• Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity
• Gravity quantized
• QSD IV : 2+1 Euclidean Quantum Gravity as a model to test 3+1 Lorentzian Quantum Gravity
• Quantum Spin Dynamics VIII. The Master Constraint
• New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis
• Chern-Simons-Matter Theory
• Separable Hilbert space in Loop Quantum Gravity
• Black hole entropy from loop quantum gravity in higher dimensions
• On the properties of the irrotational dust model
• General relativity in the radial gauge: Reduced phase space and canonical structure
• Observables for General Relativity related to geometry
• Loop quantum gravity without the Hamiltonian constraint

Cited by in corpus (14)

• (b,v)-type variables for black to white hole transitions in effective loop quantum gravity
• Quantum evolution of black hole initial data sets: Foundations
• A quantum reduction to Bianchi I models in loop quantum gravity
• An elementary introduction to loop quantum gravity
• Brief review on black hole loop quantization
• An embedding of loop quantum cosmology in (b, v) variables into a full theory context
• General relativity in the radial gauge: Reduced phase space and canonical structure
• State refinements and coarse graining in a full theory embedding of loop quantum cosmology
• 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
• Orthogonal gauge fixing of first order gravity
• Observer's observables. Residual diffeomorphisms
• On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC
• 2+1 dimensional loop quantum cosmology of Bianchi I models