Canonical quantization of (2+1)-dimensional gravity

Waelbroeck, Henri

doi:10.1103/physrevd.50.4982

Cited by 13 publications

(21 citation statements)

References 45 publications

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“…In practice, there has been fairly little work in this area, and most of the literature that does exist involves point particles rather than closed universes.’ t Hooft has emphasized that the Hamiltonian in his approach is an angle, and that time should therefore be discrete [251], in agreement with the Lorentzian spin foam analysis of Section 3.6.’ t Hooft has also found that for a particular representation of the commutation relations for a point particle in (2 + 1)-dimensional gravity, space may also be discrete [254], although it remains unclear whether these results can be generalized beyond this one special example. Criscuolo et al have examined Waelbroeck’s lattice Hamiltonian approach for the quantized torus universe [97], investigating the implication of the choice of an internal time variable, and Waelbroeck has studied the role of the mapping class group [269]. …”

Section: Quantum Gravity In 2 + 1 Dimensionsmentioning

confidence: 99%

Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe

Carlip

2005

Living Rev. Relativ.

108

120

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In three spacetime dimensions, general relativity drastically simplifies, becoming a “topological” theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2 + 1)-dimensional vacuum gravity in the setting of a spatially closed universe.

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Section: Quantum Gravity In 2 + 1 Dimensionsmentioning

confidence: 99%

Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe

Carlip

2005

Living Rev. Relativ.

108

120

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“…For an S 2 xS 1 topology, first quantisation of Dirac particles is possible. Waelbroeck has suggested a similar approach, using canonical quantisation in (2+1) dimensions [54].…”

Section: Other Approaches To Discrete Gravitymentioning

confidence: 99%

Discrete structures in gravity

Regge

Williams²

2000

Journal of Mathematical Physics

164

293

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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

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“…Quantization schemes for 2 + 1 dimensional gravity in absence of particles have been proposed in [8,9,10] and in presence of particles in [11,12,13].…”

Section: Introductionmentioning

confidence: 99%

ADM approach to 2+1 dimensional gravity

Menotti¹,

Seminara²

2000

Nuclear Physics B - Proceedings Supplements

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The canonical ADM equations are solved in terms of the conformal factor in the instantaneous York gauge. A simple derivation is given for the solution of the two body problem. A geometrical characterization is given for the apparent singularities occurring in the N -body problem and it is shown how the Garnier hamiltonian system arises in the ADM treatment by considering the time development of the conformal factor at the locations where the extrinsic curvature tensor vanishes. The equations of motion for the position of the particles and of the apparent singularities and also the time dependence of the linear residues at such singularities are given by the transformation induced by an energy momentum tensor of a conformal Liouville theory. Such an equation encodes completely the dynamics of the system.

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Canonical quantization of (2+1)-dimensional gravity

Cited by 13 publications

References 45 publications

Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe

Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe

Discrete structures in gravity

ADM approach to 2+1 dimensional gravity

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