Quantum mechanics of a point particle in -dimensional gravity
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We study the phase space structure and the quantization of a pointlike particle in (2 + 1)-dimensional gravity. By adding boundary terms to the first-order Einstein-Hilbert action, and removing all redundant gauge degrees of freedom, we arrive at a reduced action for a gravitating particle in 2 + 1 dimensions, which is invariant under Lorentz transformations and a group of generalized translations. The momentum space of the particle turns out to be the group manifold SL (2). Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime. We use the representation theory of SL (2) to investigate its structure. We find a discretization of time, and some semi-discrete structure of space. An uncertainty relation forbids a fully localized particle. The quantum dynamics is described by a discretized Klein-Gordon equation.
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(2+1)-dimensional gravity automatically displays a deformation of the Poincaré algebra associated with doubly special relativity.
“It has recently been pointed out that (2 + 1)-dimensional gravity automatically displays such a deformation [197, 18, 126].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [197]
The Symmetries section (page unavailable) says explicitly that it is not the Poincaré group itself that is deformed, but its action on phase-space variables. Its generator brackets (eq. 4.30) retain the ordinary Poincaré algebra; the deformation described is in how translations act. (LLM, conf=0.90)
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2 + 1 dimensional gravity as an exactly soluble system
Witten, E. (1988) · Nuclear Physics B · doi:10.1016/0550-3213(88)90143-5 · Cited by 2,562 · see citing works
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Research works printed but not linked: 3
- Carlip S 1993 Six ways to quantize (2 + 1)-dimensional gravity Preprint gr-qc/9305020 · arXiv
- Welling M 1997 Two particle quantum mechanics in 2 + 1 gravity using non commuting coordinates Preprint gr-qc/9703058 · arXiv
- Demichev A 1997 Physical meaning of quantum space time symmetries Preprint hep-th/9701079 · arXiv
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3D Quantum Gravity and Effective Noncommutative Quantum Field Theory
Freidel, L., & Livine, E. R. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.221301 · Cited by 298 · see citing works
Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
- Authors
- Matschull, H.-J., & Welling, M.
- Affiliations
- Utrecht University (NL)
- ORCID
- Welling, Max
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1998-10-01
- Volume
- 15
- Issue
- 10
- Pages
- 2981-3030
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2142417480
- Cited by
- 176
- References
- 1
- Citations by year
- 2026: 1 · 2025: 5 · 2024: 9 · 2023: 6 · 2022: 3 · 2021: 5 · 2020: 6 · 2019: 7 · 2018: 12 · 2017: 12 · 2016: 9 · 2015: 14 · 2014: 10 · 2013: 3 · 2012: 2
- FWCI
- 2.92
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- three-dimensional gravity, phase space structure, boundary terms, Poisson brackets, Lorentz invariance, generalized translation, uncertainty relations
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