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73
Background independent quantum gravity: a status report
, 2004
"... The goal of this article is to present an introduction to loop quantum gravity —a background independent, nonperturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to pr ..."
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Cited by 156 (3 self)
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The goal of this article is to present an introduction to loop quantum gravity —a background independent, nonperturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird’s eye view of the present status of the subject, the article should also serve as a vehicle to enter the field and explore it in detail. To aid nonexperts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the article is essentially selfcontained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the article to a reasonable size, and to avoid overwhelming nonexperts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.
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 87 (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.
The Phoenix Project: Master Constraint Programme for Loop Quantum Gravity
, 2003
"... The Hamiltonian constraint remains the major unsolved problem in Loop Quantum Gravity (LQG). Seven years ago a mathematically consistent candidate Hamiltonian constraint has been proposed but there are still several unsettled questions which concern the algebra of commutators among smeared Hamiltoni ..."
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Cited by 41 (9 self)
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The Hamiltonian constraint remains the major unsolved problem in Loop Quantum Gravity (LQG). Seven years ago a mathematically consistent candidate Hamiltonian constraint has been proposed but there are still several unsettled questions which concern the algebra of commutators among smeared Hamiltonian constraints which must be faced in order to make progress. In this paper we propose a solution to this set of problems based on the socalled Master Constraint which combines the smeared Hamiltonian constraints for all smearing functions into a single constraint. If certain mathematical conditions, which still have to be proved, hold, then not only the problems with the commutator algebra could disappear, also chances are good that one can control the solution space and the (quantum) Dirac observables of LQG. Even a decision on whether the theory has the correct classical limit and a connection with the path integral (or spin foam) formulation could be in reach. While these are exciting possibilities, we should warn the reader from the outset that, since the proposal is, to the best of our knowledge, completely new and has been barely tested in solvable models, there might be caveats which we are presently unaware of and render the whole Master Constraint Programme obsolete. Thus, this paper should really be viewed as a proposal only, rather than a presentation of hard results, which however we intend to supply in future submissions.
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.
Evolution in quantum causal histories
"... We provide a precise definition and analysis of quantum causal histories (QCH). A QCH consists of a discrete, locally finite, causal prespacetime with matrix algebras encoding the quantum structure at each event. The evolution of quantum states and observables is described by completely positive ma ..."
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Cited by 23 (3 self)
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We provide a precise definition and analysis of quantum causal histories (QCH). A QCH consists of a discrete, locally finite, causal prespacetime with matrix algebras encoding the quantum structure at each event. The evolution of quantum states and observables is described by completely positive maps between the algebras at causally related events. We show that this local description of evolution is sufficient and that unitary evolution can be recovered wherever it should actually be expected. This formalism may describe a quantum cosmology without an assumption of global hyperbolicity; it is thus more general than the WheelerDe Witt approach. The structure of a QCH is also closely related to quantum information theory and algebraic quantum field theory on a causal set.
A generalized Hamiltonian constraint operator in loop quantum gravity and its simplest Euclidean Matrix Elements, Class
 Quantum Grav
"... We study a generalized version of the Hamiltonian constraint operator in nonperturbative loop quantum gravity. The generalization is based on admitting arbitrary irreducible SU(2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental ..."
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Cited by 15 (0 self)
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We study a generalized version of the Hamiltonian constraint operator in nonperturbative loop quantum gravity. The generalization is based on admitting arbitrary irreducible SU(2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental representation is taken. This leads to a quantization ambiguity and to a family of operators with the same classical limit. We calculate the action of the Euclidean part of the generalized Hamiltonian constraint on trivalent states, using the graphical notation of TemperleyLieb recoupling theory. We discuss the relation between this generalization of the Hamiltonian constraint and crossing symmetry.
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.