Abelianized structures in spherically symmetric hypersurface deformations

Abstract

In canonical gravity, general covariance is implemented by hypersurface-deformation symmetries on phase space. The different versions of hypersurface deformations required for full covariance have complicated interplays with one another, governed by non-Abelian brackets with structure functions. For spherically symmetric space-times, it is possible to identify a certain Abelian substructure within general hypersurface deformations, which suggests a simplified realization as a Lie algebra. The generators of this substructure can be quantized more easily than full hypersurface deformations, but the symmetries they generate do not directly correspond to hypersurface deformations. The availability of consistent quantizations therefore does not guarantee general covariance or a meaningful quantum notion thereof. In addition to placing the Abelian substructure within the full context of spherically symmetric hypersurface deformation, this paper points out several subtleties relevant for attempted applications in quantized space-time structures. In particular, it follows that recent constructions by Gambini, Olmedo and Pullin in an Abelianized setting fail to address the covariance crisis of loop quantum gravity.

arXiv comment14 pages, v2: title change and more focused on hypersurface deformations

References in corpus (14)

• Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
• Loop quantization of the Schwarzschild black hole
• The Area Operator in the Spherically Symmetric Sector of Loop Quantum Gravity
• Quantum Symmetry Reduction for Diffeomorphism Invariant Theories of Connections
• Spherically Symmetric Quantum Geometry: States and Basic Operators
• Spherically Symmetric Quantum Geometry: Hamiltonian Constraint
• Covariance in models of loop quantum gravity: Spherical symmetry
• Non-marginal LTB-like models with inverse triad corrections from loop quantum gravity
• Groupoid symmetry and constraints in general relativity
• Discreteness corrections and higher spatial derivatives in effective canonical quantum gravity
• The Volume Operator in Spherically Symmetric Quantum Geometry
• Towards a quantum notion of covariance in spherically symmetric loop quantum gravity
• BV-BFV approach to General Relativity, Einstein-Hilbert action
• A Lie-Rinehart algebra in general relativity

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