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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 259 (7 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.
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 84 (19 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.
Mathematical structure of loop quantum cosmology
 Adv. Theor. Math. Phys
, 2003
"... Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and ..."
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Cited by 74 (33 self)
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Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject. 1
Asymptotic analysis of the EPRL foursimplex amplitude
, 2009
"... An asymptotic formula for a certain 4d Euclidean spin foam 4simplex amplitude is given for the limit of large spins. The analysis covers the model with Immirzi parameter less than one defined separately by Engle, Livine, Pereira and Rovelli (EPRL) and Freidel and Krasnov (FK). We are also able to a ..."
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Cited by 32 (5 self)
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An asymptotic formula for a certain 4d Euclidean spin foam 4simplex amplitude is given for the limit of large spins. The analysis covers the model with Immirzi parameter less than one defined separately by Engle, Livine, Pereira and Rovelli (EPRL) and Freidel and Krasnov (FK). We are also able to analyse the EPRL model with Immirzi parameter greater than one. The asymptotic formula has one term which is proportional to the cosine of the Regge action for gravity, and it is shown that this term is present whenever the boundary data determines a nondegenerate Euclidean geometry for the 4simplex. A scheme for resolving the phase ambiguity of the boundary data in these cases is also presented.
Cosmology of the Lifshitz universe
"... Abstract: We study the ultraviolet complete nonrelativistic theory recently proposed by Hoˇrava. After introducing a Lifshitz scalar for a general background, we analyze the cosmology of the model in Lorentzian and Euclidean signature. Vacuum solutions are found and it is argued the existence of no ..."
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Cited by 32 (4 self)
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Abstract: We study the ultraviolet complete nonrelativistic theory recently proposed by Hoˇrava. After introducing a Lifshitz scalar for a general background, we analyze the cosmology of the model in Lorentzian and Euclidean signature. Vacuum solutions are found and it is argued the existence of nonsingular bouncing profiles. We find a general qualitative agreement with both the picture of Causal Dynamical Triangulations and Quantum Einstein Gravity. However, inflation driven by a Lifshitz scalar field on a classical background might not produce a scaleinvariant
Spectral Analysis of the Volume Operator
 in Canonical Quantum General Relativity”, Diploma Thesis, Humboldt Universität zu
, 2002
"... The Volume Operator plays a crucial role in the definition of the quantum dynamics of Loop Quantum Gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the volume spectrum. While closed formulas for the matrix elements ..."
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Cited by 28 (9 self)
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The Volume Operator plays a crucial role in the definition of the quantum dynamics of Loop Quantum Gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only if one has sufficient control over the volume spectrum. While closed formulas for the matrix elements are currently available in the literature, these are complicated polynomials in 6j symbols which in turn are given in terms of Racah’s formula which is too complicated in order to perform even numerical calculations for the semiclassically important regime of large spins. Hence, so far not even numerically the spectrum could be accessed. In this article we demonstrate that by means of the Elliot – Biedenharn identity one can get rid of all the 6j symbols for any valence of the gauge invariant vertex, thus immensely reducing the computational effort. We use the resulting compact formula to study numerically the spectrum of the gauge invariant 4 – vertex. The techniques derived in this paper could be of use also for the analysis of spin – spin interaction Hamiltonians
Loop quantum gravity: An outside view
, 2005
"... We review aspects of loop quantum gravity in a pedagogical manner, with the aim of enabling a precise but critical assessment of its achievements so far. We emphasise that the offshell (‘strong’) closure of the constraint algebra is a crucial test of quantum spacetime covariance, and thereby of th ..."
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Cited by 24 (0 self)
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We review aspects of loop quantum gravity in a pedagogical manner, with the aim of enabling a precise but critical assessment of its achievements so far. We emphasise that the offshell (‘strong’) closure of the constraint algebra is a crucial test of quantum spacetime covariance, and thereby of the consistency, of the theory. Special attention is paid to the appearance of a large number of ambiguities, in particular in the formulation of the Hamiltonian constraint. Developing suitable approximation methods to establish a connection with classical gravity on the one hand, and with the physics of elementary particles on the other, remains a major challenge. Contents 1 Key questions 2