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Nelson, J. E.; Picken, Roger

doi:10.1023/a:1015574209985

Cited by 6 publications

(6 citation statements)

References 19 publications

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“…An SL(2, ℝ) matrix S is called hyperbolic, elliptic, or parabolic according to whether |tr S | is greater than, equal to, or less than 2, and the space of holonomies correspondingly splits into nine sectors. It may be shown that only the hyperbolic-hyperbolic sector corresponds to a spacetime in which the T 2 slices are spacelike [117, 119, 182, 209]. By suitable overall conjugation, the two generators of the holonomy group in this sector can then be taken to be where the are four arbitrary parameters.…”

Section: Classical Gravity In 2 + 1 Dimensionsmentioning

confidence: 99%

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

Carlip

2005

Living Rev. Relativ.

105

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: Classical Gravity In 2 + 1 Dimensionsmentioning

confidence: 99%

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

Carlip

2005

Living Rev. Relativ.

105

120

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“…In Ref. 13 we give another parametrization of the classical phase space, which is more appropriate to the present context. It consists of sectors where both matrices are diagonalizable, but also sectors where both are non-diagonalizable but can be simultaneously conjugated into upper triangular form, as well as other sectors.…”

Section: The Classical Moduli Spacementioning

confidence: 99%

“…The description of the classical phase space in terms of pairs of matrices U i is given in Ref. 13. The generalization to supergroups in the context of (2+1)-supergravity is described in Ref.…”

Section: Introductionmentioning

confidence: 99%

Quantum holonomies in (2+1)-dimensional gravity

Nelson

Picken²

2000

Physics Letters B

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We describe an approach to the quantisation of (2+1)-dimensional gravity with topology IR × T 2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q-commutation relation. Solutions of diagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matrix. Representations are constructed and a group of transformations -a quasi-modular group -which preserves this structure, is presented.

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“…The mathematical properties of just one sector have been studied in Ref. 16. In Section 2 a brief review of quantum matrices is given, whereas Section 3 discusses quantum holonomy matrices for homotopic paths, and shows how they are related by the signed area between the two paths.…”

Section: Introductionmentioning

confidence: 99%

A Quantum Goldman Bracket for Loops on Surfaces

Nelson

Picken²

2009

Int. J. Mod. Phys. A

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In the context of (2+1)-dimensional gravity, we use holonomies of constant connections which generate a q-deformed representation of the fundamental group to derive signed area phases which relate the quantum matrices assigned to homotopic loops. We use these features to determine a quantum Goldman bracket (commutator) for intersecting loops on surfaces, and discuss the resulting quantum geometry.

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Cited by 6 publications

References 19 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

Quantum holonomies in (2+1)-dimensional gravity

A Quantum Goldman Bracket for Loops on Surfaces

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