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27
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 85 (6 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 25 (3 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.
Spin foam models of Riemannian quantum gravity, available as grqc/0202017
"... Abstract. 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 m ..."
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Cited by 25 (4 self)
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Abstract. 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. 1.
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 24 (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
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 20 (5 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
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 14 (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.
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 14 (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.
The perturbative Reggecalculus regime of loop quantum gravity
 Nuclear Physics B 796 [FS] (2008) 581–621. BOJAN MOHAR AND IGOR RIVIN
"... The relation between Loop Quantum Gravity and Regge calculus has been pointed out many times in the literature. In particular the large spin asymptotics of the BarrettCrane vertex amplitude is known to be related to the Regge action. In this paper we study a semiclassical regime of Loop Quantum Gra ..."
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Cited by 3 (0 self)
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The relation between Loop Quantum Gravity and Regge calculus has been pointed out many times in the literature. In particular the large spin asymptotics of the BarrettCrane vertex amplitude is known to be related to the Regge action. In this paper we study a semiclassical regime of Loop Quantum Gravity and show that it admits an effective description in terms of perturbative areaReggecalculus. The regime of interest is identified by a class of states given by superpositions of fourvalent spin networks, peaked on large spins. As a probe of the dynamics in this regime, we compute explicitly two and threearea correlation functions at the vertex amplitude level. We find that they match with the ones computed perturbatively in areaReggecalculus with a single 4simplex, once a specific perturbative action and measure have been chosen in the Reggecalculus path integral. Correlations of other geometric operators and the existence of this regime for other models for the dynamics are briefly discussed. 1
State Sum Models for Quantum Gravity
"... Abstract. This review gives a history of the construction of quantum field theory on fourdimensional spacetime using combinatorial techniques, and recent developments of the theory towards a combinatorial construction of quantum gravity. 1. State sum models In this short review I give a brief surve ..."
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Cited by 3 (0 self)
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Abstract. This review gives a history of the construction of quantum field theory on fourdimensional spacetime using combinatorial techniques, and recent developments of the theory towards a combinatorial construction of quantum gravity. 1. State sum models In this short review I give a brief survey of the history of state sum invariants of fourmanifolds and the attempts to modify them to give models for quantum gravity. I emphasise at the outset that these are at present just models; we do not yet know how far they incorporate all the desirable features of a quantum theory of gravity. For brevity, the review will ignore the long and distinguished lowerdimensional history of these ideas. 1.1. States and weights. The general framework is as follows. Let σn be a standard nsimplex, with vertices 0, 1, 2,..., n. The state sum model requires a set of states S to be given for each simplex. These states can be thought of either as the states of a system in statistical mechanics, or as a basis set of states in quantum mechanics. This set of states is the same for any simplex of the same dimension, so one just has to specify the set of states S(σn) for each n, up to the dimension of the spacetime, n = 4. The idea is that a state on a simplex specifies a state on any one of its faces uniquely; hence there are maps ∂i: S(σn) → S(σn−1), for each i = 0,...,n, the ith map corresponding to the ith (n − 1)dimensional face (opposite the ith vertex). These satisfy some obvious relations, and the whole setup is called a simplicial set. A weight is a complex number which gives an amplitude (or Boltzmann weight) to a state. w: S(σn) → C. A state sum model uses the states and weights as the information for constructing a functional integral on a triangulated 4manifold M. A configuration on M is an assignment of states to all the simplexes in the triangulation such that the states on the faces of any simplex are given by the boundary maps ∂i. This means that the states on intersecting simplexes are related, because the common boundary data must match.