The perturbative Regge-calculus regime of loop quantum gravity

Bianchi, Eugenio; Modesto, Leonardo

doi:10.1016/j.nuclphysb.2007.12.011

Cited by 24 publications

(52 citation statements)

References 108 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This set of results strongly strengthens the relation between the quantum geometry of a spin network state and the classical simplicial geometry of piecewise-flat 3-metrics pointed out by Immirzi, by Barbieri, by Baez and by Barrett [60,58,59,61]. Notice that this relation plays a key role in the analysis of the graviton propagator in Loop Quantum Gravity [50,51,52,53,54,55].…”

Section: Semiclassical Behavioursupporting

confidence: 84%

The length operator in Loop Quantum Gravity

Bianchi

2009

Nuclear Physics B

Self Cite

123

View full text Add to dashboard Cite

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

show abstract

Section: Semiclassical Behavioursupporting

confidence: 84%

The length operator in Loop Quantum Gravity

Bianchi

2009

Nuclear Physics B

Self Cite

123

View full text Add to dashboard Cite

show abstract

“…The ratio of the Planck scale to the boundary scale will serve as a physical expansion parameter in our calculation, and we will evaluate only the lowest nontrivial order of observables in this expansion. This justifies us in using a Gaussian boundary state; higher-order corrections can be taken into account order by order in the asymptotic expansion using an improved boundary state, as discussed in [27].…”

Section: Regge Calculus and Boundary Statementioning

confidence: 99%

“…From the correlation function for lengths (34), all two-point correlation functions of local intrinsic boundary geometry observables can be obtained accurately to the lowest order. In particular the area-area correlations, which were computed for the case of a single regular simplex in [27], can be easily obtained for a general configuration. As is generally the case in semiclassical perturbation theory, the correlation functions are to the lowest order independent of the nonperturbative measure µ.…”

Section: Stationary Phase Evaluation Of Observablesmentioning

confidence: 99%

“…The graviton propagator can be extracted from the connected part of (1). This expression has been thoroughly analyzed for the Barrett-Crane [24] spin foam model [9,25,26,27,28,29] and has motivated as well the recent introduction of new models [30,31,32,33,34]. However, the analysis has been so far restricted to the case where the boundary state has support only on a single graph corresponds to the boundary of a single simplex, with the internal spin foam consisting only in the vertex amplitude that is dual to this simplex.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Semiclassical regime of Regge calculus and spin foams

Bianchi

Satz

2009

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

Recent attempts to recover the graviton propagator from spin foam models involve the use of a boundary quantum state peaked on a classical geometry. The question arises whether beyond the case of a single simplex this suffices for peaking the interior geometry in a semiclassical configuration. In this paper we explore this issue in the context of quantum Regge calculus with a general triangulation. Via a stationary phase approximation, we show that the boundary state succeeds in peaking the interior in the appropriate configuration, and that boundary correlations can be computed order by order in an asymptotic expansion. Further, we show that if we replace at each simplex the exponential of the Regge action by its cosine -as expected from the semiclassical limit of spin foam models -then the contribution from the sign-reversed terms is suppressed in the semiclassical regime and the results match those of conventional Regge calculus.

show abstract

“…In this paper, we first show that the answer to Question 2, and hence also to Question 1, is negative for every n ≥ 4. We actually found out that this has been answered previously for n = 4 in [1], where an example is given and attributed to Philip Tuckey; see also [3]. The referee has pointed out that a set of counterexamples for every n ≥ 4 (different from ours) was given by P. McMullen in [9,Sect.…”

mentioning

confidence: 67%

Simplices and Spectra of Graphs

Mohar

Rivin

2009

Discrete Comput Geom

View full text Add to dashboard Cite

In this note we show that the (n − 2)-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also show examples of families of simplices (of dimension 4 or greater) which show that the set of (n − 2)-dimensional volumes of (n − 2)-dimensional faces of a simplex do not determine its volume.

show abstract

The perturbative Regge-calculus regime of loop quantum gravity

Cited by 24 publications

References 108 publications

The length operator in Loop Quantum Gravity

The length operator in Loop Quantum Gravity

Semiclassical regime of Regge calculus and spin foams

Simplices and Spectra of Graphs

Contact Info

Product

Resources

About