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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.
Spin foam models
 Classical and Quantum Gravity
, 1998
"... While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a ‘spin foam ’ going from one spin network to another. Just as a spin network is a graph with e ..."
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Cited by 87 (2 self)
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While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a ‘spin foam ’ going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higherdimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a ‘spin foam model’, such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin networks describe ‘quantum 3geometries’, we describe how spin foams describe ‘quantum 4geometries’. We conclude by presenting a spin foam model of 4dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold.
Quantum gravity with a positive cosmological constant
, 2002
"... A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, dis ..."
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Cited by 57 (10 self)
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A quantum theory of gravity is described in the case of a positive cosmological constant in 3 + 1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the WheelerdeWitt equation corrections to the energymomentum relations for matter fields of the form E 2 = p 2 +m 2 +αlPlE 3 +... where α is a computable dimensionless constant. This may lead in the next few years to experimental tests of the theory. To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary ChernSimons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation. The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout.
Discrete approaches to quantum gravity in four dimensions
 LIVING REVIEWS IN RELATIVITY
, 1998
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Spectrum of the volume operator in quantum gravity Nucl. Phys. B 460 143; Lewandowski J 1997 Volume and quantizations Class. Quantum Grav. 14 71, Ashtekar A and Lewandowski J 1998 Quantum theory of geometry II: Volume operators Adv
 Theor. Math. Phys
, 1996
"... The volume operator is an important kinematical quantity in the nonperturbative approach to fourdimensional quantum gravity in the connection formulation. We give a general algorithm for computing its spectrum when acting on fourvalent spin network states, evaluate some of the eigenvalue formulae ..."
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Cited by 23 (9 self)
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The volume operator is an important kinematical quantity in the nonperturbative approach to fourdimensional quantum gravity in the connection formulation. We give a general algorithm for computing its spectrum when acting on fourvalent spin network states, evaluate some of the eigenvalue formulae explicitly, and discuss the role played by the Mandelstam constraints. 1 The volume operator has emerged as an important quantity in the kinematics of 3+1dimensional quantum gravity in the loop representation. It is the quantum analogue of the classical volume function, measuring the volume of threedimensional spatial regions. Although not an observable of the pure gravity theory (in the sense of commuting with
The Bekenstein bound, topological quantum field theory and pluralistic quantum cosmology
, 2008
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Properties of the volume operator in loop quantum gravity
 Properties of the Volume Operator in Loop Quantum Gravity II: Detailed Presentation. [arXiv:0706.0382
"... The properties of the Volume operator in Loop Quantum Gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. The present analysis benefits from the general simplified formula for matrix elements of the Volume operator de ..."
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Cited by 13 (1 self)
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The properties of the Volume operator in Loop Quantum Gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. The present analysis benefits from the general simplified formula for matrix elements of the Volume operator derived in [24], making it feasible to implement it on a computer as a matrix which is then diagonalized numerically. The resulting eigenvalues serve as a database to investigate the spectral properties of the volume operator. Analytical results on the spectrum at 4valent vertices are included. This is a companion paper to [25], providing details of the analysis presented there.
A discrete history of the Lorentzian path integral
, 2008
"... In these lecture notes, I describe the motivation behind a recent formulation of a nonperturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) spacetimes, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solv ..."
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Cited by 12 (3 self)
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In these lecture notes, I describe the motivation behind a recent formulation of a nonperturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) spacetimes, and give a pedagogical introduction to its main features. At the regularized, discrete level this approach solves the problems of (i) having a welldefined Wick rotation, (ii) possessing a coordinateinvariant cutoff, and (iii) leading to convergent sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d = 2 and d = 3 where continuum limits have been found. They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a nonperturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an effective regulator of quantum geometry.