Article

# Discrete Gravity Models and Loop Quantum Gravity: a Short Review

Symmetry Integrability and Geometry Methods and Applications (Impact Factor: 1.3). 04/2012; DOI: 10.3842/SIGMA.2012.052

Source: arXiv

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**ABSTRACT:**We argue that refining, coarse graining and entangling operators can be obtained from time evolution operators. This applies in particular to geometric theories, such as spin foams. We point out that this provides a construction principle for the physical vacuum in quantum gravity theories and more generally allows to construct a (cylindrically) consistent continuum limit of the theory.New Journal of Physics 11/2014; 16:123041. · 3.67 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The physical phase of causal dynamical triangulations (CDT) is known to be described by an effective, one-dimensional action in which three-volumes of the underlying foliation of the full CDT play a role of the sole degrees of freedom. Here we map this effective description onto a statistical-physics model of particles distributed on a 1d lattice, with site occupation numbers corresponding to the three-volumes. We identify the emergence of the quantum de-Sitter universe observed in CDT with the condensation transition known from similar statistical models. Our model correctly reproduces the shape of the quantum universe and allows us to analytically determine quantum corrections to the size of the universe. In the first-order approximation, these corrections lead to the widening of the CDT universe in the temporal direction. We also investigate the phase structure of the model and show that it reproduces all three phases observed in computer simulations of CDT. In addition, we predict that two other phases may exist, depending on the exact form of the discretized effective action and boundary conditions. We calculate various quantities such as the distribution of three-volumes in our model and discuss how they can be compared with CDT.Physical review D: Particles and fields 11/2012; 86(10). - [Show abstract] [Hide abstract]

**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.01/2015;

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