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Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity
General Relativity and Quantum Cosmology · Submitted 22 Aug 1997
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the phase space structure and the quantization of a pointlike particle in 2+1 dimensional gravity. | contribution 0.97 | 0.91 | 0.05 | asserted 0.77 |
| s1 | 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. | method 0.97 | 0.94 | 0.10 | asserted 1.00 |
| s2 | The momentum space of the particle turns out to be the group manifold SL(2). | result 1.00 | 0.90 | 0.14 | asserted 1.00 |
| s3 | Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime. | result 0.96 | 0.88 | 0.25 | asserted 1.00 |
| s4 | We use the representation theory of SL(2) to investigate its structure. | method 0.98 | 0.90 | 0.04 | asserted 0.90 |
| s5 | We find a discretization of time, and some semi-discrete structure of space. | result 1.00 | 0.96 | 0.22 | asserted 1.00 |
| s6 | An uncertainty relation forbids a fully localized particle. | result 0.94 | 0.87 | 0.22 | asserted 1.00 |
| s7 | The quantum dynamics is described by a discretized Klein Gordon equation. | result 0.96 | 0.90 | 0.16 | asserted 1.00 |
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Original abstract
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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References
How the literature uses it
42 arXiv papers cite this in 68 sentences
Papers citing it · 56 sentences without a confident label
Uses its method or tool 1 paper
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Following [ref] , we can write the Haar measure on $SL(2,ℝ)$ in terms of the coordinates which parametrize the group element $𝐠=u1+ξ^μγ_μ$ as [equation] where, as always, $|ξ⃗|^2=-(ξ^0)^2+|ξ|^2$.
Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity · The Haar measure · 0.92
Cites it as support 1 paper
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Additional elements of motivation for such studies are provided by the fact that $2+1$-dimensional quantum gravity, the only quantum-gravity we have managed to solve [ref] , does indeed lead to DSR-relativistic symmetries and curvature of momentum space.
Anti-de Sitter Momentum Space in 3D and 4D Quantum Gravity · Introduction · 0.97
Cites it as background 8 papers
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In three space-time dimensions Minkowski space $ℝ^2,1$ is isomorphic, as a vector space, to $𝔰𝔩(2,ℝ)$, the algebra of the Lie group $SL(2,ℝ)$.[For our readers convenience we recall the basic properties of the $SL(2,ℝ)$ group in Appendix [ref].] If $γ^μ$ is a basis of traceless $2 ×2$ matrices, given an element $𝐩 ∈𝔰𝔩(2,ℝ)$ and a vector $p⃗ ∈ℝ^2,1$ we can write [ref] [equation] Momenta are thus given by the following $𝔰𝔩(2,ℝ)$ algebra element [equation] whose determinant is $𝐩=(p^0)^2-|p|^2$.
Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity · Particles as conical defects in $2+1$ gravity · 0.85
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In [ref] , quantum mechanics of the massive particles coupled with three dimensional Einstein gravity has been considered.
Massive particles coupled with 2+1 dimensional gravity and noncommutative field theory · Introduction · 0.97
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Further it is argued that the presence of the mass scale has an effect similar to the one well known in 2+1 gravity [ref] , namely that the momentum space becomes `curved'[The idea of a curved momentum space appeared already in Refs. [ref] and [ref] , and was later developed by the Russian group (see Ref. [ref] and references therein).], i.e., starts having a non-trivial geometry, with the characteristic scale $κ$ [ref] .
Relative Locality in Curved Space-time · Introduction · 0.81
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