Knot Theory and Quantum Gravity

Rovelli, Carlo; Smolin, Lee

doi:10.1103/physrevlett.61.1155

Cited by 407 publications

(529 citation statements)

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“…Now we move on to the quantum theory [19,20]. The essential assumption in Loop Quantum Gravity is that the mathematically well-defined operators acting on the Hilbert space are the holonomy 2 of the connection along a curve e, h e [A], and the smearing of the two-form E i = E a i (x) ε abc dx b ∧ dx c over a 1 The following notational conventions are adopted: a, b, .…”

Section: Quantization Of 3-geometric Observablesmentioning

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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Section: Quantization Of 3-geometric Observablesmentioning

confidence: 99%

“…Here we follow [43]. We represent the Wigner metric and the three-valent intertwining tensor by an oriented line and by a node with three links oriented counter-clockwise 19 , respectively:…”

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confidence: 99%

The length operator in Loop Quantum Gravity

Bianchi

2009

Nuclear Physics B

123

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“…The first is the canonical approach, where one tries to formulate the model in a background independent way, i.e. to give space time at Planckian scales a polymer-like structure, similar to 'spin networks', and from these (rather abstract) building blocks to construct the interactions [1] and the observable universe. The second approach is the one in which the quantum gravitational interactions are represented as a 'stochastic medium', which gives space time non-trivial optical properties ('space-time foam') [2].…”

Section: Introductionmentioning

confidence: 99%

Testing a (Stringy) Model of Quantum Gravity

Mavromatos

2001

Dark Matter in Astro- And Particle Physics

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Abstract. I discuss a specific model of space-time foam, inspired by the modern nonperturbative approach to string theory (D-branes). The model views our world as a three brane, intersecting with D-particles that represent stringy quantum gravity effects, which can be real or virtual. In this picture, matter is represented generically by (closed or open) strings on the D3 brane propagating in such a background. Scattering of the (matter) strings off the D-particles causes recoil of the latter, which in turn results in a distortion of the surrounding space-time fluid and the formation of (microscopic, i.e. Planckian size) horizons around the defects. As a mean-field result, the dispersion relation of the various particle excitations is modified, leading to non-trivial optical properties of the space time, for instance a non-trivial refractive index for the case of photons or other massless probes. Such models make falsifiable predictions, that may be tested experimentally in the foreseeable future. I describe a few such tests, ranging from observations of light from distant gamma-ray-bursters and ultra high energy cosmic rays, to tests using gravity-wave interferometric devices and terrestrial particle physics experiments involving, for instance, neutral kaons.

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“…This is a central question for practically all theoretical attempts, including noncommutative geometry [20,21], supersymmetry and superstrings theories [43,112], for which the compactification scale is close to the Planck scale, and particularly for the theory of quantum gravity. Indeed, the development of loop quantum gravity by Rovelli and Smolin [114] led to the conclusion that the Planck scale could be a quantized minimal scale in Nature, involving also a quantization of surfaces and volumes [115].…”

Section: Special Scale-relativitymentioning

confidence: 99%

Scale Relativity and Fractal Space-Time: Theory and Applications

Nottale

2010

Found Sci

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In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations.In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-periodic law for species punctuated evolution, human development and society evolution), to Earth sciences (log-periodic deceleration of the rate of California earthquakes and of Sichuan earthquake replicas, critical law for the arctic sea ice extent) and tentative applications to system biology.

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Knot Theory and Quantum Gravity

Cited by 407 publications

References 3 publications

The length operator in Loop Quantum Gravity

The length operator in Loop Quantum Gravity

Testing a (Stringy) Model of Quantum Gravity

Scale Relativity and Fractal Space-Time: Theory and Applications

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