Discrete Gravity Models and Loop Quantum Gravity: a Short Review

Symmetry Integrability and Geometry Methods and Applications (Impact Factor: 1.25). 04/2012; 8. DOI: 10.3842/SIGMA.2012.052
Source: arXiv


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

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    • "Its quantum states of geometry are the spin network states and can be understood as the quantization of twisted geometries. These are a generalization of Regge geometries allowing for torsion (which encodes extrinsic curvature in the context of loop quantum gravity) introduced by Freidel & Speziale [3] and then further developed as the natural geometrical interpretation of loop quantum gravity [4] [5] [6] [7]. They can be understood as patchwork of flat polyhedra, individually embedded in flat 3d space, glued together through area matching conditions across their triangular faces but without face shape matching. "
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    ABSTRACT: We investigate the generalization of loop gravity's twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space R^3. One then glues them allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Lambda, and in particular that deforming the loop gravity phase space with real parameter q>0 would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in SU(2) associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in SB(2,C)). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.
    Preview · Article · Jan 2015 · Classical and Quantum Gravity
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    • "Moreover, the perturbative sum over Feynman diagrams coincides with the definition of quantum gravity given by the (Euclidean) Dynamical triangulations approach [10], after appropriate identification of their respective parameter sets. When one enriches the combinatorics of tensor models with the group-theoretic data suggested by Loop Quantum Gravity [9], Spin Foam models [16] and simplicial geometry [11], one obtains (Tensorial) Group Field Theories: proper field theories, with richer state spaces (with generic states being superpositions of spin networks) and quantum amplitudes, given by simplicial path integrals and spin foam models. It is these richer field theories, building up on the understanding of quantum geometry obtained in loop quantum gravity, that we believe offer the most promising candidates for a complete quantum theory of gravity. "
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    ABSTRACT: The definition of a double-scaling limit represents an important goal in the development of tensor models. We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the IID tensor models. We show that the radius of convergence of the NLO series coincides with that of the leading order melonic sector. Meanwhile, the value of the susceptibility exponent at NLO is 3/2, signaling a departure from the leading order behaviour. Both pieces of information provide clues for a non-trivial double-scaling limit, for which we put forward some precise conjecture.
    Full-text · Article · Apr 2013 · New Journal of Physics
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    • "To start with, the introduction of spinor variables allowed a compact reformulation of the loop gravity phase space [2] [3] [4] [5] [6], with a clear geometrical interpretation as " twisted geometries " generalizing the discrete Regge geometries. This became particularly relevant for the construction and interpretation of spinfoam models when analyzing the hierarchy of constraints to impose on arbitrary discrete space-time geometries in order to implement a proper quantum version of general relativity [7]. Furthermore, following the generalization of these spinor variables to twistor networks allowing to describe a Lorentz connection [8] [9], these spinor techniques (or actually upgraded to twistor techniques) allowed to explore and better understand the phase space structure underlying the discrete path integral defining the spinfoam amplitudes [10] [11] [12]. "
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    ABSTRACT: We construct the holonomy-flux operator algebra in the recently developed spinor formulation of loop gravity. We show that, when restricting to SU(2)-gauge invariant operators, the familiar grasping and Wilson loop operators are written as composite operators built from the gauge-invariant `generalized ladder operators' recently introduced in the U(N) approach to intertwiners and spin networks. We comment on quantization ambiguities that appear in the definition of the holonomy operator and use these ambiguities as a toy model to test a class of quantization ambiguities which is present in the standard regularization and definition of the Hamiltonian constraint operator in loop quantum gravity.
    Preview · Article · Feb 2013 · Physical review D: Particles and fields
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