Simplification of the spectral analysis of the volume operator in loop quantum gravity

Brunnemann, Johannes; Thiemann, Thomas

doi:10.1088/0264-9381/23/4/014

Cited by 100 publications

(250 citation statements)

References 42 publications

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“…Unfortunately, these steps are all but straightforward to carry out in practice [116]. The analysis can be simplified by applying the operator identity [115,117] …”

Section: The Volume Operatormentioning

confidence: 99%

Loop quantum gravity: an outside view

Nicolai

Peeters

Zamaklar

2005

Class. Quantum Grav.

215

319

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Abstract:We review aspects of loop quantum gravity in a pedagogical manner, with the aim of enabling a precise but critical assessment of its achievements so far. We emphasise that the off-shell ('strong') closure of the constraint algebra is a crucial test of quantum space-time covariance, and thereby of the consistency, of the theory. Special attention is paid to the appearance of a large number of ambiguities, in particular in the formulation of the Hamiltonian constraint. Developing suitable approximation methods to establish a connection with classical gravity on the one hand, and with the physics of elementary particles on the other, remains a major challenge.

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“…Unfortunately, these steps are all but straightforward to carry out in practice [116]. The analysis can be simplified by applying the operator identity [115,117] …”

Section: The Volume Operatormentioning

confidence: 99%

Loop quantum gravity: an outside view

Nicolai

Peeters

Zamaklar

2005

Class. Quantum Grav.

215

319

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“…Therefore its matrix elements are non-trivial only in the intertwiner sector and can be computed using standard recoupling techniques [26,28,29]. In formulae we have that…”

Section: Quantization Of the Volumementioning

confidence: 99%

“…Iαβ and the local inverse-volume operator V −1 I as prescribed by equation (29). Both of them, the two-hand operator and the local inverse-volume operator, admit a dual description in terms of nodes and links of the graph of a spin network state.…”

Section: Quantization: the 'Elementary' Length Operatormentioning

confidence: 99%

“…The coefficients a(k, j 1 , j 2 , j 3 , j 4 ) are real and admit the following explicit expression 23 [28,29] …”

Section: Matrix Elements Of the Length Operator: Four Valent Nodesmentioning

confidence: 99%

“…The non-zero eigenvalues always appear in pairs with opposite sign. A zero eigenvalue is present only when the dimension of the intertwiner vector space is odd [28,29].…”

Section: Matrix Elements Of the Length Operator: Four Valent Nodesmentioning

confidence: 99%

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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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Kinematics: Spatial Atoms

Bojowald

2011

Quantum Cosmology

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Simplification of the spectral analysis of the volume operator in loop quantum gravity

Cited by 100 publications

References 42 publications

Loop quantum gravity: an outside view

Loop quantum gravity: an outside view

The length operator in Loop Quantum Gravity

Kinematics: Spatial Atoms

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