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27
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 127 (8 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
A Lorentzian signature model for quantum general relativity,” grqc/9904025
"... Abstract. We give a relativistic spin network model for quantum gravity based on the Lorentz group and its qdeformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state ..."
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Cited by 107 (7 self)
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Abstract. We give a relativistic spin network model for quantum gravity based on the Lorentz group and its qdeformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the fourdimensional rotation group previously studied in [1], grqc/9709028. As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the ‘10J ’ symbol needed in our model has a finite value. 1.
Integrability For Relativistic Spin Networks
"... The evaluation of relativistic spin networks plays a fundamental role in the BarrettCrane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decom ..."
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Cited by 35 (4 self)
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The evaluation of relativistic spin networks plays a fundamental role in the BarrettCrane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L² functions on threedimensional hyperbolic space. To `evaluate' such a spin network we must do an integral; if this integral converges we say the spin network is `integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4simplex (the complete graph on 5 vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model.
On Relativistic Spin Network Vertices
 J. Math. Phys
, 1999
"... Barrett and Crane have proposed a model of simplicial Euclidean quantum gravity in which a central role is played by a class of Spin(4) spin networks called “relativistic spin networks ” which satisfy a series of physically motivated constraints. Here a proof is presented that demonstrates that the ..."
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Cited by 31 (0 self)
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Barrett and Crane have proposed a model of simplicial Euclidean quantum gravity in which a central role is played by a class of Spin(4) spin networks called “relativistic spin networks ” which satisfy a series of physically motivated constraints. Here a proof is presented that demonstrates that the intertwiner of a vertex of such a spin network is uniquely determined, up to normalization, by the representations on the incident edges and the constraints. Moreover, the constraints, which were formulated for four valent spin networks only, are extended to networks of arbitrary valence, and the generalized relativistic spin networks proposed by Yetter are shown to form the entire solution set (mod normalization) of the extended constraints. Finally, using the extended constraints, the BarrettCrane model is generalized to arbitrary polyhedral complexes (instead of just simplicial complexes) representing spacetime. It is explained how this model, like the BarretCrane model can be derived from BF theory by restricting the sum over histories to ones in which the left handed and right handed areas of any 2surface are equal. It is known that the solutions of classical Euclidean GR form a branch of the stationary points of the BF action with respect to variations preserving this condition. 1
Spin foam models of Riemannian quantum gravity
, 2002
"... Using numerical calculations, we compare three versions of the Barrett– Crane model of 4dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many tria ..."
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Cited by 27 (4 self)
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Using numerical calculations, we compare three versions of the Barrett– Crane model of 4dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many triangulated 4manifolds. In the version with modified face and edge amplitudes due to Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We also describe a new version which appears to have a convergent partition function without drastic spinzero dominance. Finally, after a general discussion of how to extract physics from spin foam models, we discuss the implications of convergence or divergence of the partition function for other aspects of a spin foam model.
Geometrical measurements in threedimensional quantum gravity
 J. Phys. A: Math. Gen. 18 Suppl
"... A set of observables is described for the topological quantum field theory which describes quantum gravity in three spacetime dimensions with positive signature and positive cosmological constant. The simplest examples measure the distances between points, giving spectra and probabilities which hav ..."
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Cited by 25 (7 self)
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A set of observables is described for the topological quantum field theory which describes quantum gravity in three spacetime dimensions with positive signature and positive cosmological constant. The simplest examples measure the distances between points, giving spectra and probabilities which have a geometrical interpretation. The observables are related to the evaluation of relativistic spin networks by a Fourier transform. 1
Coarse graining methods for spin net and spin foam models,” New J. Phys. 14, 035008 (2012) [arXiv:1109.4927 [grqc
 Holonomy Spin Foam Models: Definition and Coarse Graining,” Phys. Rev. D 87, 044048 (2013) [arXiv:1208.3388 [grqc
"... 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 20 (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
Asymptotics of 10j symbols
 Classical Quantum Gravity
"... Abstract. The Riemannian 10j symbols are spin networks that assign an amplitude to each 4simplex in the BarrettCrane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4simplex, and Barrett and Williams have shown that one contribution to its asy ..."
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Cited by 18 (2 self)
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Abstract. The Riemannian 10j symbols are spin networks that assign an amplitude to each 4simplex in the BarrettCrane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all nondegenerate 4simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a ‘degenerate spin network’, where the rotation group SO(4) is replaced by the Euclidean group of isometries of R 3. We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulas in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the Riemannian ones. 1.
Positivity of Spin Foam Amplitudes
 Class. Quantum Grav
"... The amplitude for a spin foam in the BarrettCrane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are al ..."
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Cited by 13 (3 self)
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The amplitude for a spin foam in the BarrettCrane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. This means one can use the Metropolis algorithm to compute expectation values of observables in the Riemannian BarrettCrane model, as in statistical mechanics, even though this theory is based on a realtime (e iS ) rather than imaginarytime (e S ) path integral. Our proof uses the fact that when the Riemannian 10j symbols are nonzero, their sign is positive or negative depending on whether the sum of the ten spins is an integer or halfinteger. For the product of 10j symbols appearing in the amplitude for a closed spin foam, these signs cancel. We conclude with some numerical evidence suggesting that the Lorentzian 10j symbols are always nonnegative.