Discrete Gravity Models and Loop Quantum Gravity: a Short Review

Abstract

We review the relation between Loop Quantum Gravity on a fixed graph and discrete models of gravity. We compare Regge and twisted geometries, and discuss discrete actions based on twisted geometries and on the discretization of the Plebanski action. We discuss the role of discrete geometries in the spin foam formalism, with particular attention to the definition of the simplicity constraints.

v2: many amendments, matches published version

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• Laplacians on discrete and quantum geometries
• Geometrizing the Klein-Gordon and Dirac equations in Doubly Special Relativity
• Closure constraints for hyperbolic tetrahedra
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• Regge calculus models of the closed vacuum $\Lambda$-FLRW universe
• Renormalization of group field theories for quantum gravity: new scaling results and some suggestions
• The Construction of Spin Foam Vertex Amplitudes
• Quantum Surface and Intertwiner Dynamics in Loop Quantum Gravity
• Modification of the mean free path of very high energy photons due to a relativistic deformed kinematics
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• Time delays, choice of energy-momentum variables and relative locality in doubly special relativity
• Hamiltonian Renormalisation V: Free Vector Bosons
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• A new perspective on Doubly Special Relativity
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• Geometry from local flatness in Lorentzian spin foam theories
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• Spinfoam tunneling of quantum geometries in angle variables