The length operator in Loop Quantum Gravity

Abstract

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

Curvature operator for loop quantum gravity

“…In LQG we have three proposals for length operator [22][23][24]. Since our approach to construct a scalar curvature operator is using the dual picture, we choose Bianchi's operator [23] which is constructed based on the same dual picture of quantum geometry.…”

“…Since our approach to construct a scalar curvature operator is using the dual picture, we choose Bianchi's operator [23] which is constructed based on the same dual picture of quantum geometry. Here we summarize the construction contained in [23].…”

“…Since our approach to construct a scalar curvature operator is using the dual picture, we choose Bianchi's operator [23] which is constructed based on the same dual picture of quantum geometry. Here we summarize the construction contained in [23]. The first step in this construction is an external regularization of the classical expression of the length of a curve (10).…”

“…note that there exist two versions of the volume operator [33,34] in the literature; the one presented in (22) and used in [23] is the Rovelli-Smolin version [33] which is consistent with the external regularization scheme. In principle one could use here the version [34] even if this is based on a different regularization procedure.…”

“…lengths and angles. Operators of length [22][23][24] and angle [25][26][27] are available in LQG and Spinfoams and by using a suitable regularization procedure for the classical quantities we can implement directly the integral of the Ricci scalar as an operator acting on spin-network states and in this way settle the first step to bypass many of the complications appearing in the Lorentzian constraint. The considerations due to the change of regularization procedure bring us close to the perspective of [10] and make our proposal intriguing also for the spinfoam formalism.…”