Spin Networks in Gauge Theory

Baez, John C.

doi:10.1006/aima.1996.0012

Cited by 216 publications

(268 citation statements)

References 11 publications

Supporting

Mentioning

263

Contrasting

Unclassified

Order By: Relevance

“…At the level of cylindrical functions this amounts to restricting attention to the class of functions invariant under SU (2) transformations at the nodes of the graph. Spin network states [21,22] provide an orthonormal basis of K 0 . They are defined in the following way.…”

Section: Quantization Of 3-geometric Observablesmentioning

confidence: 99%

The length operator in Loop Quantum Gravity

Bianchi

2009

Nuclear Physics B

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: Quantization Of 3-geometric Observablesmentioning

confidence: 99%

The length operator in Loop Quantum Gravity

Bianchi

2009

Nuclear Physics B

123

View full text Add to dashboard Cite

show abstract

“…As mentioned in the section on 3-dimensional gravity, spin networks have played an important role in calculations of invariants of 3-manifolds, and in loop quantum gravity, where they provide a gauge-invariant basis of states [75,93]. If we wish to describe space-time by this type of method, we need, as we have already remarked, an extension of the concept of spin networks.…”

Section: E Spin Foammentioning

confidence: 99%

Discrete structures in gravity

Regge

Williams²

2000

Journal of Mathematical Physics

164

293

View full text Add to dashboard Cite

Discrete approaches to gravity, both classical and quantum, are reviewed briefly, with emphasis on the method using piecewise-linear spaces. Models of 3-dimensional quantum gravity involving 6j-symbols are then described, and progress in generalising these models to four dimensions is discussed, as is the relationship of these models in both three and four dimensions to topological theories. Finally, the repercussions of the generalisations are explored for the original formulation of discrete gravity using edge-length variables. I Introduction to discrete gravity A Basic formalismThe original motivation for the development of a discrete formalism for gravity [1] arose from a number of problems with the continuum formulation of general relativity. These included the difficulty of solving Einstein's equations for general systems without a large degree of symmetry, the problems of representing complicated topologies and the need for considerable geometric insight and capacity for visualisation. It turned out, as we shall see, that the discretisation scheme to be described not only helped with these problems but also found a vital rôle in numerical relativity and in attempts at a formulation of quantum gravity.The related branches of mathematics which found their application to physics in this formulation of gravity are those of piecewise-linear spaces and topology and the geometric notion of intrinsic curvature on polyhedra. The immediate aim was to develop an approach to general relativity which avoided the 1

show abstract

“…The following proposition has been proved by J. Baez [1]. For the sake of completeness and because we find it illuminating, we give a short proof of Theorem 4.1.…”

Section: Spin Networkmentioning

confidence: 88%

Wilson loops in the light of spin networks

LEVY¹

2004

Journal of Geometry and Physics

View full text Add to dashboard Cite

If G is any finite product of orthogonal, unitary and symplectic matrix groups, then Wilson loops generate a dense subalgebra of continuous observables on the configuration space of lattice gauge theory with structure group G. If G is orthogonal, unitary or symplectic, then Wilson loops associated to the natural representation of G are enough.This extends a result of A. Sengupta [7]. In particular, our approach includes the case of even orthogonal groups. IntroductionOn a compact Lie group, the Peter-Weyl theorem asserts that the characters of irreducible representations generate a dense subalgebra of continuous functions invariant by adjunction. In lattice gauge theory, configuration spaces are powers of a Lie group on which another power of the same group acts, according to the geometry of a given graph and in a way which extends the adjoint action of the group on itself. Peter-Weyl theorem can be adapted to this situation and the functions that play the role of the characters are called spin networks. Despite the fact that spin networks were introduced about forty years ago in a physical context 1 , their importance in lattice gauge theory has been recognized rather recently [1]. In the mean time, another set of functions, easier to define, has been used as the standard set of observables: Wilson loops. However, it is not clear at all a priori that this set is complete, that is, that Wilson loops generate a dense subalgebra of continuous invariant functions on the configuration space. A. Sengupta has proved in [7] that it is true when the group is a product of odd orthogonal, unitary (and symplectic) groups. In this paper, an approach similar to that of Sengupta but with a little more classical invariant theory combined with the use of spin networks allows us to add even orthogonal groups to the list and, hopefully, to clarify the argument. * IRMA -7, rue René Descartes -F-67084 Strasbourg Cedex 1 R. Penrose introduced them for the purposes of quantization of the geometry of space. See [8] for a historical account.

show abstract

Spin Networks in Gauge Theory

Cited by 216 publications

References 11 publications

The length operator in Loop Quantum Gravity

The length operator in Loop Quantum Gravity

Discrete structures in gravity

Wilson loops in the light of spin networks

Contact Info

Product

Resources

About