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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
Inverse scale factor in isotropic quantum geometry
, 2001
"... The inverse scale factor, which in classical cosmological models diverges at the singularity, is quantized in isotropic models of loop quantum cosmology by using techniques which have been developed in quantum geometry for a quantization of general relativity. This procedure results in a bounded ope ..."
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Cited by 56 (38 self)
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The inverse scale factor, which in classical cosmological models diverges at the singularity, is quantized in isotropic models of loop quantum cosmology by using techniques which have been developed in quantum geometry for a quantization of general relativity. This procedure results in a bounded operator which is diagonalizable simultaneously with the volume operator and whose eigenvalues are determined explicitly. For large scale factors (in fact, up to a scale factor slightly above the Planck length) the eigenvalues are close to the classical expectation, whereas below the Planck length there are large deviations leading to a nondiverging behavior of the inverse scale factor even though the scale factor has vanishing eigenvalues. This is a first indication that the classical singularity is better behaved in loop quantum cosmology.
The Semiclassical Limit of Loop Quantum Cosmology
, 2001
"... The continuum and semiclassical limits of isotropic, spatially flat loop quantum cosmology are discussed, with an emphasis on the role played by the Barbero–Immirzi parameter γ in controlling spacetime discreteness. In this way, standard quantum cosmology is shown to be the simultaneous limit γ → 0 ..."
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Cited by 28 (20 self)
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The continuum and semiclassical limits of isotropic, spatially flat loop quantum cosmology are discussed, with an emphasis on the role played by the Barbero–Immirzi parameter γ in controlling spacetime discreteness. In this way, standard quantum cosmology is shown to be the simultaneous limit γ → 0, j → ∞ of loop quantum cosmology. Here, j is a label of the volume eigenvalues, and the simultaneous limit is technically the same as the classical limit � → 0, l → ∞ of angular momentum in quantum mechanics. Possible lessons for semiclassical states at the dynamical level in the full theory of quantum geometry are mentioned.
Towards the QFT on Curved Spacetime Limit of QGR. I: A General Scheme
"... In this article and the companion paper [1] we address the question of how one might obtain the semiclassical limit of ordinary matter quantum fields (QFT) propagating on curved spacetimes (CST) from full fledged Quantum General Relativity (QGR), starting from first principles. We stress that we do ..."
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Cited by 26 (6 self)
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In this article and the companion paper [1] we address the question of how one might obtain the semiclassical limit of ordinary matter quantum fields (QFT) propagating on curved spacetimes (CST) from full fledged Quantum General Relativity (QGR), starting from first principles. We stress that we do not claim to have a satisfactory answer to this question, rather our intention is to ignite a discussion by displaying the problems that have to be solved when carrying out such a program. In the first paper of this series of two we propose a general scheme of logical steps that one has to take in order to arrive at such a limit. We discuss the technical and conceptual problems that arise in doing so and how they can be solved in principle. As to be expected, completely new issues arise due to the fact that QGR is a background independent theory. For instance, fundamentally the notion of a photon involves not only the Maxwell quantum field but also the metric operator – in a sense, there is no photon vacuum state but a “photon vacuum operator”! Such problems have, to the best of our knowledge, not been discussed in the literature before, we are facing squarely one aspect of the deep conceptual difference between a
The Dark Side of a Patchwork Universe
, 2007
"... While observational cosmology has recently progressed fast, it revealed a serious dilemma called dark energy: an unknown source of exotic energy with negative pressure driving a current accelerating phase of the universe. All attempts so far to find a convincing theoretical explanation have failed, ..."
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Cited by 21 (19 self)
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While observational cosmology has recently progressed fast, it revealed a serious dilemma called dark energy: an unknown source of exotic energy with negative pressure driving a current accelerating phase of the universe. All attempts so far to find a convincing theoretical explanation have failed, so that one of the last hopes is the yet to be developed quantum theory of gravity. In this article, loop quantum gravity is considered as a candidate, with an emphasis on properties which might play a role for the dark energy problem. Its basic feature is the discrete structure of space, often associated with quantum theories of gravity on general grounds. This gives rise to welldefined matter Hamiltonian operators and thus sheds light on conceptual questions related to the cosmological constant problem. It also implies typical quantum geometry effects which, from a more phenomenological point of view, may result in dark energy. In particular the latter scenario allows several nontrivial tests which can be made more precise by detailed observations in combination with a quantitative study of numerical quantum gravity. If the speculative possibility of a loop quantum gravitational origin of dark energy turns out to be realized, a program as outlined here will help to hammer out our ideas for a quantum theory of gravity, and at the same time allow predictions for the distant future of our universe.
Semiclassical analysis of the Loop Quantum Gravity volume operator: I. Flux Coherent States
, 2008
"... The volume operator plays a pivotal role for the quantum dynamics of Loop Quantum Gravity (LQG), both in the full theory and in truncated models adapted to cosmological situations coined Loop Quantum Cosmology (LQC). It is therefore crucial to check whether its semiclassical limit coincides with the ..."
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Cited by 19 (4 self)
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The volume operator plays a pivotal role for the quantum dynamics of Loop Quantum Gravity (LQG), both in the full theory and in truncated models adapted to cosmological situations coined Loop Quantum Cosmology (LQC). It is therefore crucial to check whether its semiclassical limit coincides with the classical volume operator plus quantum corrections. In the present article we investigate this question by generalizing and employing previously defined coherent states for LQG which derive from a cylindrically consistently defined complexifier operator which is the quantization of a known classical function. These coherent states are not normalizable due to the non separability of the LQG Hilbert space but they define uniquely define cut – off states depending on a finite graph. The result of our analysis is that the expectation value of the volume operator with respect to coherent states depending on a graph with only n−valent verticies reproduces its classical value at the phase space point at which the coherent state is peaked only if n = 6. In other words, the semiclassical sector of LQG defined by those states is described by graphs with cubic topology! This has some bearing on current
Degenerate Configurations, Singularities and the NonAbelian Nature of Loop Quantum Gravity
 Class. Quantum Grav
, 2006
"... Degenerate geometrical configurations in quantum gravity are important to understand if the fate of classical singularities is to be revealed. However, not all degenerate configurations arise on an equal footing, and one must take into account dynamical aspects when interpreting results: While there ..."
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Cited by 18 (18 self)
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Degenerate geometrical configurations in quantum gravity are important to understand if the fate of classical singularities is to be revealed. However, not all degenerate configurations arise on an equal footing, and one must take into account dynamical aspects when interpreting results: While there are many degenerate spatial metrics, not all of them are approached along the dynamical evolution of general relativity or a candidate theory for quantum gravity. For loop quantum gravity, relevant properties and steps in an analysis are summarized and evaluated critically with the currently available information, also elucidating the role of degrees of freedom captured in the sector provided by loop quantum cosmology. This allows an outlook on how singularity removal might be analyzed in a general setting and also in the full theory. The general mechanism of loop quantum cosmology will be shown to be insensitive to recently observed unbounded behavior of inverse volume in the full theory. Moreover, significant features of this unboundedness are not a consequence of inhomogeneities but of nonAbelian effects which can also be included in homogeneous models. 1
Loop quantum cosmology, boundary proposals, and inflation
 Phys. Rev. D
"... Loop quantum cosmology of the closed isotropic model is studied with a special emphasis on a comparison with traditional results obtained in the Wheeler–DeWitt approach. This includes the relation of the dynamical initial conditions with boundary conditions such as the noboundary or the tunneling p ..."
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Cited by 18 (12 self)
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Loop quantum cosmology of the closed isotropic model is studied with a special emphasis on a comparison with traditional results obtained in the Wheeler–DeWitt approach. This includes the relation of the dynamical initial conditions with boundary conditions such as the noboundary or the tunneling proposal and a discussion of inflation from quantum cosmology. 1
Polymer and Fock representations for a Scalar field
, 2002
"... In loop quantum gravity, matter fields can have support only on the ‘polymerlike’ excitations of quantum geometry, and their algebras of observables and Hilbert spaces of states can not refer to a classical, background geometry. Therefore, to adequately handle the matter sector, one has to address ..."
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Cited by 17 (1 self)
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In loop quantum gravity, matter fields can have support only on the ‘polymerlike’ excitations of quantum geometry, and their algebras of observables and Hilbert spaces of states can not refer to a classical, background geometry. Therefore, to adequately handle the matter sector, one has to address two issues already at the kinematic level. First, one has to construct the appropriate background independent operator algebras and Hilbert spaces. Second, to make contact with low energy physics, one has to relate this ‘polymer description’ of matter fields to the standard Fock description in Minkowski space. While this task has been completed for gauge fields, important gaps remained in the treatment of scalar fields. The purpose of this letter is to fill these gaps.
Fundamental structure of loop quantum gravity
 Int. J. Mod. Phys
, 2007
"... In recent twenty years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. The aim of loop quantum gravity is to construct a mathematically rigorous, background independent, nonperturbative quantum theory for Loren ..."
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Cited by 14 (6 self)
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In recent twenty years, loop quantum gravity, a background independent approach to unify general relativity and quantum mechanics, has been widely investigated. The aim of loop quantum gravity is to construct a mathematically rigorous, background independent, nonperturbative quantum theory for Lorentzian gravitational field on four dimensional manifold. In the approach, the principles of quantum mechanics are combined with those of general relativity naturally. Such a combination gives us a picture of, so called, quantum Riemannian geometry, which is discrete at fundamental scale. Imposing the quantum constraints in analogy from the classical ones, the quantum dynamics of gravity is being studied as one of the most important issues in loop quantum gravity. On the other hand, the semiclassical analysis is being carried out to test the classical limit of the quantum theory. In this review, the fundamental structure of loop quantum gravity is