A note on the secondary simplicity constraints in loop quantum gravity

Abstract

A debate has appeared in the literature on loop quantum gravity and spin foams, over whether the secondary simplicity constraints, reducing the connection to be Levi-Civita, should imply the shape matching conditions, reducing twisted geometries to Regge geometries. We address the question using a simple model with a flat dynamics, in which secondary simplicity constraints arise from a dynamical preservation of the primary ones. We find that shape matching conditions arise, thus providing support to an affirmative question. The origin of these extra conditions is to be found in the different graph localisation of the Hamiltonian and primary simplicity constraints. Our results are consistent with previous claims by Dittrich and Ryan, and extend their validity to Lorentzian signature and arbitrary cellular decompositions. We show in particular how the (gauge-invariant version of the) twist angle xi featuring in twisted geometries equals on-shell the Regge dihedral angle multiplied by the Immirzi parameter, thus recovering the discrete extrinsic geometry from the Ashtekar-Barbero holonomy. Finally, we confirm that flatness implies both the Levi-Civita and the shape-matching conditions using twisted geometries and a 4-dimensional version of the vertex condition appearing in 't Hooft's polygon model.

20 pages. v2 many improvements: slightly different Hamiltonian, more details on the canonical analysis and on the extension to arbitrary graphs, new section added. Main conclusion unchanged, but made more precise

References in corpus (14)

• LQG vertex with finite Immirzi parameter
• A new spinfoam vertex for quantum gravity
• The loop-quantum-gravity vertex-amplitude
• Polyhedra in loop quantum gravity
• Algebraic Quantum Gravity (AQG) I. Conceptual Setup
• Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
• Area-angle variables for general relativity
• U(N) tools for Loop Quantum Gravity: The Return of the Spinor
• Generalized Spinfoams
• Null twisted geometries
• Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces
• New action for simplicial gravity in four dimensions
• Classification of constraints and degrees of freedom for quadratic discrete actions
• Loop quantum gravity, twistors, and some perspectives on the problem of time

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• Twisted Geometries Coherent States for Loop Quantum Gravity
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