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Quantum Gravity
, 2004
"... We describe the basic assumptions and key results of loop quantum gravity, which is a background independent approach to quantum gravity. The emphasis is on the basic physical principles and how one deduces predictions from them, at a level suitable for physicists in other areas such as string theor ..."
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Cited by 363 (9 self)
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We describe the basic assumptions and key results of loop quantum gravity, which is a background independent approach to quantum gravity. The emphasis is on the basic physical principles and how one deduces predictions from them, at a level suitable for physicists in other areas such as string theory, cosmology, particle physics, astrophysics and condensed matter physics. No details are given, but references are provided to guide the interested reader to the literature. The present state of knowledge is summarized in a list of 35 key results on topics including the hamiltonian and path integral quantizations, coupling to matter, extensions to supergravity and higher dimensional theories, as well as applications to black holes, cosmology and Plank scale phenomenology. We describe the near term prospects for observational tests of quantum theories of gravity and the expectations that loop quantum gravity may provide predictions for their outcomes. Finally, we provide answers to frequently asked questions and a list of key open problems.
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 162 (5 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.
Relativistic Spin Networks and Quantum Gravity
 J. Math Phys
, 1998
"... Abstract. Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) × SU(2). Relativistic quantum spins are related to the geometry of the 2dimensional faces of a 4simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geome ..."
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Cited by 134 (15 self)
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Abstract. Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) × SU(2). Relativistic quantum spins are related to the geometry of the 2dimensional faces of a 4simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3simplex. This leads us to suggest that there may be a 4dimensional state sum model for quantum gravity based on relativistic spin networks which parallels the construction of 3dimensional quantum gravity from ordinary spin networks.
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 48 (9 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.
Quantum Geometry of Isolated Horizons and Black Hole Entropy
, 2000
"... Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a nonperturbative quantization of the sector of general relativity, coupled to matter, admitting nonrotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polym ..."
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Cited by 44 (3 self)
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Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a nonperturbative quantization of the sector of general relativity, coupled to matter, admitting nonrotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic geometry of the horizon is then described by the quantum ChernSimons theory of a U(1) connection on a punctured 2sphere, the horizon. Subtle mathematical features of the quantum ChernSimons theory turn out to be important for the existence of a coherent quantum theory of the horizon geometry. Heuristically, the intrinsic geometry is flat everywhere except at the punctures. The distributional curvature of the U(1) connection at the punctures gives rise to quantized deficit angles which account for the overall curvature. For macroscopic black holes, the logarithm of the number of these horizon microstates is proportional to the area, irrespective of the values of (nongravitational) charges. Thus, the black hole entropy can be accounted for entirely by the quantum states of the horizon geometry. Our analysis is applicable to all nonrotating black holes, including the astrophysically interesting ones which are very far from extremality. Furthermore, cosmological horizons (to which statistical mechanical considerations are known to apply) are naturally incorporated. An effort has been made to make the paper selfcontained by including short reviews of the background material.
Quantum geometry with intrinsic local causality
, 1997
"... The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact twodimensional surfaces. The space of states of the theory is the direct sum of the ..."
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Cited by 39 (17 self)
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The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact twodimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group Gq over all compact (finite genus) oriented 2surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism.
Structural Issues in Quantum Gravity
, 1995
"... A discursive, nontechnical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of bo ..."
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Cited by 25 (2 self)
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A discursive, nontechnical, analysis is made of some of the basic issues that arise in almost any approach to quantum gravity, and of how these issues stand in relation to recent developments in the field. Specific topics include the applicability of the conceptual and mathematical structures of both classical general relativity and standard quantum theory. This discussion is preceded by a short history of the last twentyfive years of research in quantum gravity, and concludes with speculations on what a future theory might look like.
A holographic formulation of quantum general relativity”, Phys
 Rev. D
"... We show that there is a sector of quantum general relativity, in the Lorentzian signature case, which may be expressed in a completely holographic formulation in terms of states and operators defined on a finite boundary. The space of boundary states is built out of the conformal blocks of SU(2)L ⊕ ..."
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Cited by 22 (11 self)
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We show that there is a sector of quantum general relativity, in the Lorentzian signature case, which may be expressed in a completely holographic formulation in terms of states and operators defined on a finite boundary. The space of boundary states is built out of the conformal blocks of SU(2)L ⊕ SU(2)R, WZW field theory on the npunctured sphere, where n is related to the area of the boundary. The Bekenstein bound is explicitly satisfied. These results are based on a new lagrangian and hamiltonian formulation of general relativity based on a constrained Sp(4) topological field theory. The hamiltonian formalism is polynomial, and also leftright symmetric. The quantization uses balanced SU(2)L ⊕SU(2)R spin networks and so justifies the state sum model of Barrett and Crane. By extending the formalism to Osp(4/N) a holographic formulation of extended supergravity is obtained, as will be described in detail in a subsequent paper.
Lectures on loop quantum gravity
 LECT. NOTES PHYS
, 2003
"... ... the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures. Background independence means that this is a nonper ..."
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Cited by 18 (1 self)
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... the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures. Background independence means that this is a nonperturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein’s radical perception that gravity is geometry. Minimality here means that one explores the logical consequences of bringing together the two fundamental principles of modern physics, namely general covariance and quantum theory, without adding any experimentally unverified additional structures such as extra dimensions, extra symmetries or extra particle content beyond the standard model. While this is a very conservative approach and thus maybe not very attractive to many researchers, it has the advantage that pushing the theory to its logical frontiers will undoubtedly either result in a successful theory or derive exactly which extra structures are required, if necessary. Or put even more radically, it may show which basic principles of physics have to be given up and must be replaced by more fundamental ones.
Supersymmetric spin networks and quantum supergravity
 Phys.Rev
, 2000
"... We define supersymmetric spin networks, which provide a complete set of gauge invariant states for supergravity and supersymmetric gauge theories. The particular case of Osp(1/2) is studied in detail and applied to the nonperturbative quantization of supergravity. The supersymmetric extension of th ..."
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Cited by 17 (9 self)
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We define supersymmetric spin networks, which provide a complete set of gauge invariant states for supergravity and supersymmetric gauge theories. The particular case of Osp(1/2) is studied in detail and applied to the nonperturbative quantization of supergravity. The supersymmetric extension of the area operator is defined and partly diagonalized. The spectrum is discrete as in quantum general relativity, and the two cases could be distinguished by measurements of quantum geometry. 1