Comment on ‘New variables for 1 + 1 dimensional gravity’ (2010 Class. Quantum Grav . 27 025002)

Quantum Holonomies in (2+1)-Dimensional Gravity

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 co...

Lessons from (2+1)-dimensional quantum gravity

Proposals that quantum gravity gives rise to non-commutative spacetime geometry and deformations of Poincaré symmetry are examined in the context of (2+1)dimensional quantum gravity. The results are expressed in five lessons, which summarise how the gravitational constant, Planck’s constant and the cosmological constant enter the non-commutative and non-cocommutative structures arising in (2+1)...

Time in (2+1)-Dimensional Quantum Gravity

General relativity in three spacetime dimensions is used to explore three approaches to the “problem of time” in quantum gravity: the internal Schrödinger approach with mean extrinsic curvature as a time variable, the Wheeler-DeWitt equation, and covariant canonical quantization with “evolving constants of motion.” (To appear in Proc. of the Lanczos Centenary Conference, Raleigh, NC, December 1...

Geometric Structures and Loop Variables in (2+1)-Dimensional Gravity

Ashtekar’s connection representation for general relativity [1, 2] and the closely related loop variable approach [3] have generated a good deal of excitement over the past few years. While it is too early to make firm predictions, there seems to be some real hope that these new variables will allow the construction of a consistent nonperturbative quantum theory of gravity. Some important progr...

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