Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

Lewandowski, Jerzy; Okołów, Andrzej; Sahlmann, Hanno; Thiemann, Thomas

doi:10.1007/s00220-006-0100-7

Cited by 288 publications

(549 citation statements)

References 24 publications

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“…Nevertheless, one can derive all basic operators given by holonomies and fluxes of the symmetric model from corresponding ones in the full theory. This is sufficient to derive the quantum representation of models from the unique one in the full theory [61,62]. Since many properties of loop quantizations, including those discussed here, follow already from basic aspects the derivation of the basic representation is a crucial step.…”

Section: Symmetry Reduction: From Inhomogeneity To Homogeneous Modelsmentioning

confidence: 99%

Loop quantum cosmology and inhomogeneities

2006

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Inhomogeneities are introduced in loop quantum cosmology using regular lattice states, with a kinematical arena similar to that in homogeneous models considered earlier. The framework is intended to encapsulate crucial features of background independent quantizations in a setting accessible to explicit calculations of perturbations on a cosmological background. It is used here only for qualitative insights but can be extended with further more detailed input. One can thus see how several parameters occuring in homogeneous models appear from an inhomogeneous point of view. Their physical roles in several cases then become much clearer, often making previously unnatural choices of values look more natural by providing alternative physical roles. This also illustrates general properties of symmetry reduction at the quantum level and the roles played by inhomogeneities. Moreover, the constructions suggest a picture for gravitons and other metric modes as collective excitations in a discrete theory, and lead to the possibility of quantum gravity corrections in large universes.

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Section: Symmetry Reduction: From Inhomogeneity To Homogeneous Modelsmentioning

confidence: 99%

Loop quantum cosmology and inhomogeneities

2006

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“…Now let's focus on a cell R I which contains a node of Γ. In this case, some further adaptation of the regularized expression (13) to the graph Γ is required. The point x I and the three curves introduced by T ijk xI in the definition of the regularized volume are adapted to the graph Γ in the following way: -the point x I in (11) is chosen to coincide with the position of the node, -the three curves γ 1 xI σ , γ 2 xI σ and γ 3 xI σ are adapted to three of the links of Γ originating at the node contained in the cell R I .…”

Section: Quantization Of the Volumementioning

confidence: 99%

The length operator in Loop Quantum Gravity

Bianchi

2009

Nuclear Physics B

123

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The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity -the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.Keywords: quantum geometry; spin network states; Planck-scale discreteness PACS: 04.60.Pp; 04.60.Nc A remarkable feature of the loop approach [1, 2, 3] to the problem of quantum gravity [4] is the prediction of a quantum discreteness of space at the Planck scale. Such discreteness manifests itself in the analysis of the spectrum of geometric operators describing the volume of a region of space [5,6] or the area of a surface separating two such regions [5,7]. In this paper we introduce a new operator -the length operator -study its properties and show that it has a discrete spectrum and an appropriate semiclassical behaviour. For a different attempt to introduce a length operator in Loop Quantum Gravity, see Thiemann's paper [8]. For some remarks about why it is difficult to introduce a length operator is Loop Quantum Gravity see the review [9]. In the following we describe the picture of quantum geometry coming from Loop Quantum Gravity and the role played by the length in this picture (section 1), we recall the standard procedure used in Loop Quantum Gravity when introducing an operator corresponding to a given classical geometrical quantity (section 2), we point out the difficulties to overcome in order to introduce the length operator (section 3.1), discuss the strategy that we follow (sections 3.2-3.3) and finally present our results in sections 4 and 5.1 The dual picture of quantum geometryIn Loop Quantum Gravity, the state of the 3-geometry can be given in terms of a linear superposition of spin network states. Such spin network states consist of a graph embedded in a 3-manifold and a coloring of its edges and its nodes in terms of SU(2) irreducible representations and of SU(2) intertwiners. Thanks to the existence of a volume operator and an area operator, the following dual picture of the quantum geometry of a spin network state is available (see sections 1.2.2 and 6.7 of [1] for a detailed * e.bianchi@sns.it 1

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“…Quite surprisingly, the requirement that the representation be cyclic with respect to a state which is invariant under the action of the group of (piecewise analytic) diffeomorphisms on M singles out a unique irreducible representation. This result was recently established for a by Lewandowski, Oko lów, Sahlmann and Thiemann [8], and for W by Fleischhack [9]. It is the quantum geometry analog to the seminal results by Segal and others that characterized the Fock vacuum in Minkowskian field theories.…”

Section: Quantum Riemannian Geometrymentioning

confidence: 55%

Gravity, Geometry and the Quantum

Ashtekar¹,

Oluseyi²

2009

AIP Conference Proceedings

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After a brief introduction, basic ideas of the quantum Riemannian geometry underlying loop quantum gravity are summarized. To illustrate physical ramifications of quantum geometry, the framework is then applied to homogeneous isotropic cosmology. Quantum geometry effects are shown to replace the big bang by a big bounce. Thus, quantum physics does not stop at the big-bang singularity. Rather there is a pre-big-bang branch joined to the current post-big-bang branch by a 'quantum bridge'. Furthermore, thanks to the background independence of loop quantum gravity, evolution is deterministic across the bridge.

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Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

Cited by 288 publications

References 24 publications

Loop quantum cosmology and inhomogeneities

Loop quantum cosmology and inhomogeneities

The length operator in Loop Quantum Gravity

Gravity, Geometry and the Quantum

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