Fractal Structure of Loop Quantum Gravity

In this paper we have calculated the spectral dimension of loop quantum gravity (LQG) using simple arguments coming from the area spectrum at different length scales. We have obtained that the spectral dimension of the spatial section runs from 2 to 3, across a 1.5 phase, when the energy of a probe scalar field decrees from high to low energy. We have calculated the spectral dimension of the space-time also using results from spin-foam models, obtaining a 2-dimensional effective manifold at hight energy. Our result is consistent with other two approach to non perturbative quantum gravity: causal dynamical triangulation and asymptotic safety quantum gravity.Comment: 5 pages, 5 figure

A spin foam model for general Lorentzian 4-geometries

We derive simplicity constraints for the quantization of general Lorentzian 4-geometries. Our method is based on the correspondence between coherent states and classical bivectors and the minimization of associated uncertainties. For spacelike geometries, this scheme agrees with the master constraint method of the model by Engle, Pereira, Rovelli and Livine (EPRL). When it is applied to general Lorentzian geometries, we obtain new constraints that include the EPRL constraints as a special case. They imply a discrete area spectrum for both spacelike and timelike surfaces. We use these constraints to define a spin foam model for general Lorentzian 4-geometries.Comment: 27 pages, 1 figure; v4: published versio

Holomorphic Factorization for a Quantum Tetrahedron

We provide a holomorphic description of the Hilbert space H(j_1,..,j_n) of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for the decomposition of identity in H(j_1,..,j_n). Interestingly, the integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. Our results provide a new interpretation for this quantity as being, in the limit of large conformal dimensions, the exponential of the Kahler potential of the symplectic manifold whose quantization gives H(j_1,..,j_n). For the case n=4, the symplectic manifold in question has the interpretation of the space of "shapes" of a geometric tetrahedron with fixed face areas, and our results provide a description for the quantum tetrahedron in terms of holomorphic coherent states. We describe how the holomorphic intertwiners are related to the usual real ones by computing their overlap. The semi-classical analysis of these overlap coefficients in the case of large spins allows us to obtain an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron. Our results are of direct relevance for the subjects of loop quantum gravity and spin foams, but also add an interesting new twist to the story of the bulk/boundary correspondence.Comment: 45 pages; published versio

A finiteness bound for the EPRL/FK spin foam model

We show that the EPRL/FK spin foam model of quantum gravity has an absolutely convergent partition function if the vertex amplitude is divided by an appropriate power $p$ of the product of dimensions of the vertex spins. This power is independent of the spin foam 2-complex and we find that $p>2$ insures the convergence of the state sum. Determining the convergence of the state sum for the values $0 \le p \le 2$ requires the knowledge of the large-spin asymptotics of the vertex amplitude in the cases when some of the vertex spins are large and other are small.Comment: v6: published versio

Coupling gauge theory to spinfoam 3d quantum gravity

We construct a spinfoam model for Yang-Mills theory coupled to quantum gravity in three dimensional riemannian spacetime. We define the partition function of the coupled system as a power series in g_0^2 G that can be evaluated order by order using grasping rules and the recoupling theory. With respect to previous attempts in the literature, this model assigns the dynamical variables of gravity and Yang-Mills theory to the same simplices of the spinfoam, and it thus provides transition amplitudes for the spin network states of the canonical theory. For SU(2) Yang-Mills theory we show explicitly that the partition function has a semiclassical limit given by the Regge discretization of the classical Yang-Mills action.Comment: 18 page

Colored Group Field Theory

Group field theories are higher dimensional generalizations of matrix models. Their Feynman graphs are fat and in addition to vertices, edges and faces, they also contain higher dimensional cells, called bubbles. In this paper, we propose a new, fermionic Group Field Theory, posessing a color symmetry, and take the first steps in a systematic study of the topological properties of its graphs. Unlike its bosonic counterpart, the bubbles of the Feynman graphs of this theory are well defined and readily identified. We prove that this graphs are combinatorial cellular complexes. We define and study the cellular homology of this graphs. Furthermore we define a homotopy transformation appropriate to this graphs. Finally, the amplitude of the Feynman graphs is shown to be related to the fundamental group of the cellular complex

Second-order amplitudes in loop quantum gravity

We explore some second-order amplitudes in loop quantum gravity. In particular, we compute some second-order contributions to diagonal components of the graviton propagator in the large distance limit, using the old version of the Barrett-Crane vertex amplitude. We illustrate the geometry associated to these terms. We find some peculiar phenomena in the large distance behavior of these amplitudes, related with the geometry of the generalized triangulations dual to the Feynman graphs of the corresponding group field theory. In particular, we point out a possible further difficulty with the old Barrett-Crane vertex: it appears to lead to flatness instead of Ricci-flatness, at least in some situations. The observation raises the question whether this difficulty remains with the new version of the vertex.Comment: 22 pages, 18 figure

Cosmological Plebanski theory

We consider the cosmological symmetry reduction of the Plebanski action as a toy-model to explore, in this simple framework, some issues related to loop quantum gravity and spin-foam models. We make the classical analysis of the model and perform both path integral and canonical quantizations. As for the full theory, the reduced model admits two types of classical solutions: topological and gravitational ones. The quantization mixes these two solutions, which prevents the model to be equivalent to standard quantum cosmology. Furthermore, the topological solution dominates at the classical limit. We also study the effect of an Immirzi parameter in the model.Comment: 20 page

Asymptotics of 4d spin foam models

We study the asymptotic properties of four-simplex amplitudes for various four-dimensional spin foam models. We investigate the semi-classical limit of the Ooguri, Euclidean and Lorentzian EPRL models using coherent states for the boundary data. For some classes of geometrical boundary data, the asymptotic formulae are given, in all three cases, by simple functions of the Regge action for the four-simplex geometry.Comment: 10 pages, Proceedings for the 2nd Corfu summer school and workshop on quantum gravity and quantum geometry, talk given by Winston J. Fairbair

Disordered locality in loop quantum gravity states

We show that loop quantum gravity suffers from a potential problem with non-locality, coming from a mismatch between micro-locality, as defined by the combinatorial structures of their microscopic states, and macro-locality, defined by the metric which emerges from the low energy limit. As a result, the low energy limit may suffer from a disordered locality characterized by identifications of far away points. We argue that if such defects in locality are rare enough they will be difficult to detect.Comment: 11 pages, 4 figures, revision with extended discussion of result