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Spin Foam Models for Quantum Gravity
, 2008
"... In this article we review the present status of the spin foam formulation of nonperturbative (background independent) quantum gravity. The article is divided in two parts. In the first part we present a general introduction to the main ideas emphasizing their motivations from various perspectives. R ..."
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Cited by 123 (7 self)
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In this article we review the present status of the spin foam formulation of nonperturbative (background independent) quantum gravity. The article is divided in two parts. In the first part we present a general introduction to the main ideas emphasizing their motivations from various perspectives. Riemannian 3dimensional gravity is used as a simple example to illustrate conceptual issues and the main goals of the approach. The main features of the various existing models for 4dimensional gravity are also presented here. We conclude with a discussion of important questions to be addressed in four dimensions (gauge invariance, discretization independence, etc.). In the second part we concentrate on the definition of the BarrettCrane model. We present the main results obtained in this framework from a critical perspective. Finally we review the combinatorial formulation of spin foam models based on the dual group field theory technology. We present the BarrettCrane model in this framework and review the finiteness results obtained for both its Riemannian as well
Coarse graining methods for spin net and spin foam models
 HOLONOMY SPIN FOAM MODELS: DEFINITION AND COARSE GRAINING,” PHYS. REV. D 87, 044048 (2013) [ARXIV:1208.3388 [GRQC
, 2011
"... We undertake first steps in making a class of discrete models of quantum gravity, spin foams, accessible to a large scale analysis by numerical and computational methods. In particular, we apply MigdalKadanoff and Tensor Network Renormalization schemes to spin net and spin foam models based on fini ..."
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Cited by 19 (11 self)
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We undertake first steps in making a class of discrete models of quantum gravity, spin foams, accessible to a large scale analysis by numerical and computational methods. In particular, we apply MigdalKadanoff and Tensor Network Renormalization schemes to spin net and spin foam models based on finite Abelian groups and introduce ‘cutoff models’ to probe the fate of gauge symmetries under various such approximated renormalization group flows. For the Tensor Network Renormalization analysis, a new Gauß constraint preserving algorithm is introduced to improve numerical stability and aid physical interpretation. We
Renormalization of discrete models without background
 Nucl. Phys
"... Conventional renormalization methods in statistical physics and lattice quantum field theory assume a flat metric background. We outline here a generalization of such methods to models on discretized spaces without metric background. Cellular decompositions play the role of discretizations. The grou ..."
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Cited by 15 (2 self)
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Conventional renormalization methods in statistical physics and lattice quantum field theory assume a flat metric background. We outline here a generalization of such methods to models on discretized spaces without metric background. Cellular decompositions play the role of discretizations. The group of scale transformations is replaced by the groupoid of changes of cellular decompositions. We introduce cellular moves which generate this groupoid and allow to define a renormalization groupoid flow. We proceed to test our approach on several models. Quantum BF theory is the simplest example as it is almost topological and the renormalization almost trivial. More interesting is generalized lattice gauge theory for which a qualitative picture of the renormalization groupoid flow can be given. This is confirmed by the exact renormalization in dimension two.
Implementing causality in the spin foam quantum geometry
 Nucl. Phys. B
"... We analyse the classical and quantum geometry of the BarrettCrane spin foam model for four dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be con ..."
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Cited by 11 (0 self)
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We analyse the classical and quantum geometry of the BarrettCrane spin foam model for four dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be consistently implemented in this framework, and construct causal transiton amplitudes between quantum gravity states, i.e. realising in the spin foam context the Feynman propagator between states. The resulting causal spin foam model can be seen as a path integral quantization of Lorentzian first order Regge calculus, and represents a link between several approaches to quantum gravity as canonical loop quantum gravity, sumoverhistories formulations, dynamical triangulations and causal sets. In particular, we show how the resulting model can be rephrased within the framework of quantum causal sets (or histories).
Geometric spin foams, YangMills theory and backgroundindependent models
, 2005
"... We review the dual transformation from pure lattice gauge theory to spin foam models with an emphasis on a geometric viewpoint. This allows us to give a simple dual formulation of SU(N) YangMills theory, where spin foam surfaces are weighted with the exponentiated area. In the case of gravity, we i ..."
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Cited by 8 (2 self)
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We review the dual transformation from pure lattice gauge theory to spin foam models with an emphasis on a geometric viewpoint. This allows us to give a simple dual formulation of SU(N) YangMills theory, where spin foam surfaces are weighted with the exponentiated area. In the case of gravity, we introduce a symmetry condition which demands that the amplitude of an individual spin foam depends only on its geometric properties and not on the lattice on which it is defined. For models that have this property, we define a new sum over abstract spin foams that is independent of any choice of lattice or triangulation. We show that a version of the BarrettCrane model satisfies our symmetry requirement.
Exact duality transformations for sigma models and gauge theories
, 2003
"... We present an exact duality transformation in the framework of Statistical Mechanics for various lattice models with nonAbelian global or local symmetries. The transformation applies to sigma models with variables in a compact Lie group G with global G × Gsymmetry (the chiral model) and with varia ..."
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Cited by 7 (5 self)
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We present an exact duality transformation in the framework of Statistical Mechanics for various lattice models with nonAbelian global or local symmetries. The transformation applies to sigma models with variables in a compact Lie group G with global G × Gsymmetry (the chiral model) and with variables in coset spaces G/H and a global N Gsymmetry (for example, the nonlinear O(N) or models) in any dimension d ≥ 1. It is also available for lattice gauge theories with local gauge symmetry in dimensions d ≥ 2 and for the models obtained from minimally coupling a sigma model of the type mentioned above to a gauge theory. The duality transformation maps the strong coupling regime of the original model to the weak coupling regime of the dual model. Transformations are available for the partition function, for expectation values of fundamental variables (correlators and generalized Wilson loops) and for expectation values in the dual model which correspond in the original formulation to certain ratios of partition functions (free energies of dislocations, vortices or monopoles). Whereas the original models are formulated in terms of compact Lie groups G and H, coset spaces G/H and integrals over them, the configurations of the dual model are given in terms of representations and intertwiners of G and H. They are spin networks and spin foams. The partition function of the dual model describes the group theoretic aspects of the strong coupling expansion in a closed form.
Positivity of relativistic spin network evaluations
, 2002
"... Let G be a compact Lie group. Using suitable normalization conventions, we show that the evaluation of G ×Gsymmetric spin networks is nonnegative whenever the edges are labeled by representations of the form V ⊗ V ∗ where V is a representation of G, and the intertwiners are generalizations of the ..."
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Cited by 3 (2 self)
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Let G be a compact Lie group. Using suitable normalization conventions, we show that the evaluation of G ×Gsymmetric spin networks is nonnegative whenever the edges are labeled by representations of the form V ⊗ V ∗ where V is a representation of G, and the intertwiners are generalizations of the Barrett–Crane intertwiner. This includes in particular the relativistic spin networks with symmetry group Spin(4) or SO(4). We also present a counterexample, using the finite group S3, to the stronger conjecture that all spin network evaluations are nonnegative as long as they can be written using only group integrations and index contractions. This counterexample applies in particular to the product of five 6jsymbols which appears in the spin foam model of the S3symmetric BFtheory on the twocomplex dual to a triangulation of the sphere S 3 using five tetrahedra. We show that this product is negative real for a particular assignment of representations to the edges. PACS: 04.60.Nc key words: Spin network, spin network evaluations, spin foam model
On the causal Barrett–Crane model: measure, coupling constant, Wick rotation, symmetries and observables
, 2002
"... We discuss various features and details of two versions of the Barrett–Crane spin foam model of quantum gravity, first of the Spin(4)symmetric Riemannian model and second of the SL(2,�)symmetric Lorentzian version in which all tetrahedra are spacelike. Recently, Livine and Oriti proposed to intro ..."
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Cited by 2 (0 self)
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We discuss various features and details of two versions of the Barrett–Crane spin foam model of quantum gravity, first of the Spin(4)symmetric Riemannian model and second of the SL(2,�)symmetric Lorentzian version in which all tetrahedra are spacelike. Recently, Livine and Oriti proposed to introduce a causal structure into the Lorentzian Barrett–Crane model from which one can construct a path integral that corresponds to the causal (Feynman) propagator. We show how to obtain convergent integrals for the 10jsymbols and how a dimensionless constant can be introduced into the model. We propose a ‘Wick rotation ’ which turns the rapidly oscillating complex amplitudes of the Feynman path integral into positive real and bounded weights. This construction does not yet have the status of a theorem, but it can be used as an alternative definition of the propagator and makes the causal model accessible by standard numerical simulation algorithms. In addition, we identify the local symmetries of the models and show how their foursimplex amplitudes can be reexpressed in terms of the ordinary relativistic 10jsymbols. Finally, motivated by possible numerical simulations, we express the matrix elements that are defined by the model, in terms of the continuous connection variables and determine the most general observable in the connection picture. Everything is done on a fixed twocomplex. PACS: 04.60.Nc