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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
Loop quantum cosmology
, 2006
"... Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e. the fact that the backward evolution of a classical spacetime inevitably comes to an end after a ..."
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Cited by 48 (11 self)
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Quantum gravity is expected to be necessary in order to understand situations where classical general relativity breaks down. In particular in cosmology one has to deal with initial singularities, i.e. the fact that the backward evolution of a classical spacetime inevitably comes to an end after a finite amount of proper time. This presents a breakdown of the classical picture and requires an extended theory for a meaningful description. Since small length scales and high curvatures are involved, quantum effects must play a role. Not only the singularity itself but also the surrounding spacetime is then modified. One particular realization is loop quantum cosmology, an application of loop quantum gravity to homogeneous systems, which removes classical singularities. Its implications can be studied at different levels. Main effects are introduced into effective classical equations which allow to avoid interpretational problems of quantum theory. They give rise to new kinds of early universe phenomenology with applications to inflation and cyclic models. To resolve classical singularities and to understand the structure of geometry around them, the quantum description is necessary. Classical evolution is then replaced by a difference equation for a wave function which allows to extend spacetime beyond classical singularities. One main question is how these homogeneous scenarios are related to full loop quantum gravity, which can be dealt with at the level of distributional symmetric states. Finally, the new structure of spacetime arising in loop quantum gravity and its application to cosmology sheds new light on more general issues such as time.
Quantization ambiguities in isotropic quantum geometry
, 2002
"... Some typical quantization ambiguities of quantum geometry are studied within isotropic models. Since this allows explicit computations of operators and their spectra, one can investigate the effects of ambiguities in a quantitative manner. It is shown that those ambiguities do not affect the fate of ..."
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Cited by 44 (29 self)
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Some typical quantization ambiguities of quantum geometry are studied within isotropic models. Since this allows explicit computations of operators and their spectra, one can investigate the effects of ambiguities in a quantitative manner. It is shown that those ambiguities do not affect the fate of the classical singularity, demonstrating that the absence of a singularity in loop quantum cosmology is a robust implication of the general quantization scheme. The calculations also allow conclusions about modified operators in the full theory. In particular, using holonomies in a nonfundamental representation of SU(2) to quantize connection components turns out to lead to significant corrections to classical behavior at macroscopic volume for large values of the spin of the chosen representation. 1
Spherically Symmetric Quantum Geometry: States and Basic
 Operators, Class. Quantum Grav
"... Abstract The kinematical setting of spherically symmetric quantum geometry, derived from the full theory of loop quantum gravity, is developed. This extends previous studies of homogeneous models to inhomogeneous ones where interesting field theory aspects arise. A comparison between a reduced quant ..."
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Cited by 35 (21 self)
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Abstract The kinematical setting of spherically symmetric quantum geometry, derived from the full theory of loop quantum gravity, is developed. This extends previous studies of homogeneous models to inhomogeneous ones where interesting field theory aspects arise. A comparison between a reduced quantization and a derivation of the model from the full theory is presented in detail, with an emphasis on the resulting quantum representation. Similar concepts for EinsteinRosen waves are discussed briefly.
Spin foam models of Riemannian quantum gravity
, 2002
"... 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 tria ..."
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Cited by 27 (4 self)
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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.
Lattice refining loop quantum cosmology, anisotropic models and stability
, 2007
"... A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In genera ..."
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Cited by 24 (19 self)
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A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new feature of dynamical difference equations which may not have constant stepsize, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum gravity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed.
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
Relating covariant and canonical approaches to triangulated models of quantum gravity, available as grqc/0110026
"... Abstract. In this paper explore the relation between covariant and canonical approaches to quantum gravity and BF theory. We will focus on the dynamical triangulation and spinfoam models, which have in common that they can be defined in terms of sums over spacetime triangulations. Our aim is to sh ..."
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Cited by 10 (0 self)
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Abstract. In this paper explore the relation between covariant and canonical approaches to quantum gravity and BF theory. We will focus on the dynamical triangulation and spinfoam models, which have in common that they can be defined in terms of sums over spacetime triangulations. Our aim is to show how we can recover these covariant models from a canonical framework by providing two regularisations of the projector onto the kernel of the Hamiltonian constraint. This link is important for the understanding of the dynamics of quantum gravity. In particular, we will see how in the simplest dynamical triangulations model we can recover the Hamiltonian constraint via our definition of the projector. Our discussion of spinfoam models will show how the elementary spinnetwork moves in loop quantum gravity, which were originally assumed to describe the Hamiltonian constraint action, are in fact related to the timeevolution generated by the constraint. We also show that the Immirzi parameter is important for the understanding of a continuum limit of the theory. 1.
Perturbative Degrees of Freedom in Loop Quantum Gravity: Anisotropies
, 2005
"... The relation between an isotropic and an anisotropic model in loop quantum cosmology is discussed in detail, comparing the strict symmetry reduction with a perturbative implementation of symmetry. While the latter cannot be done in a canonical manner, it allows to consider the dynamics including the ..."
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Cited by 9 (7 self)
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The relation between an isotropic and an anisotropic model in loop quantum cosmology is discussed in detail, comparing the strict symmetry reduction with a perturbative implementation of symmetry. While the latter cannot be done in a canonical manner, it allows to consider the dynamics including the role of small nonsymmetric degrees of freedom for the symmetric evolution. This serves as a model for the general situation of perturbative degrees of freedom in a background independent quantization such as loop quantum gravity, and for the more complicated addition of perturbative inhomogeneities. While being crucial for cosmological phenomenology, it is shown that perturbative nonsymmetric degrees of freedom do not allow definitive conclusions for the singularity issue and in such a situation could even lead to wrong claims.
The volume operator for spin networks with planar or cylindrical symmetry. [grqc/0511005
"... This paper constructs a kinematic basis for spin networks with planar or cylindrical symmetry, by exploiting the fact that the basis elements are representations of an O(3) subgroup of O(4). The action of the volume operator on this basis gives a difference equation for the eigenvalues and eigenvect ..."
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Cited by 7 (3 self)
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This paper constructs a kinematic basis for spin networks with planar or cylindrical symmetry, by exploiting the fact that the basis elements are representations of an O(3) subgroup of O(4). The action of the volume operator on this basis gives a difference equation for the eigenvalues and eigenvectors of the volume operator. For basis elements of low spin, the difference equation can be solved readily on a computer, yielding the eigenvalues and eigenvectors. For higher spins, I solve for the eigenvalues using a WKBJ method. This paper considers only the case where the gravitational wave can have both polarizations. The single polarization case is considered in a separate paper. PACS categories: 04.60, 04.30 I