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Quantization of Lorentzian 3d Gravity by Partial Gauge Fixing
General Relativity and Quantum Cosmology · Submitted 24 Apr 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We make use of this fact to show that, for positive cosmological constant, the Lorentzian theory can be partially gauge fixed and reduced to an SU(2) Chern-Simons theory.
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The abstract, sentence by sentence
D = 2+1 gravity with a cosmological constant has been shown by Bonzom and Livine to present a Barbero-Immirzi like ambiguity depending on a parameter. We make use of this fact to show that, for positive cosmological constant, the Lorentzian theory can be partially gauge fixed and reduced to an SU(2) Chern-Simons theory. We then review the already known quantization of the latter in the framework of Loop Quantization for the case of space being topogically a cylinder. We finally construct, in the same setting, a quantum observable which, although non-trivial at the quantum level, corresponds to a null classical quantity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | D = 2+1 gravity with a cosmological constant has been shown by Bonzom and Livine to present a Barbero-Immirzi like ambiguity depending on a parameter. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s1 | We make use of this fact to show that, for positive cosmological constant, the Lorentzian theory can be partially gauge fixed and reduced to an SU(2) Chern-Simons theory. | contribution 0.65 | 0.94 | 0.63 | asserted 1.00 |
| s2 | We then review the already known quantization of the latter in the framework of Loop Quantization for the case of space being topogically a cylinder. | background 0.59 | 0.11 | 0.06 | asserted 0.96 |
| s3 | We finally construct, in the same setting, a quantum observable which, although non-trivial at the quantum level, corresponds to a null classical quantity. | result 0.71 | 0.94 | 0.36 | asserted 1.00 |
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The paper
Original abstract
D = 2+1 gravity with a cosmological constant has been shown by Bonzom and Livine to present a Barbero-Immirzi like ambiguity depending on a parameter. We make use of this fact to show that, for positive cosmological constant, the Lorentzian theory can be partially gauge fixed and reduced to an SU(2) Chern-Simons theory. We then review the already known quantization of the latter in the framework of Loop Quantization for the case of space being topogically a cylinder. We finally construct, in the same setting, a quantum observable which, although non-trivial at the quantum level, corresponds to a null classical quantity.
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How the literature uses it
1 arXiv paper cite this in 9 sentences
Papers citing it · 5 sentences without a confident label
Builds on it 1 paper
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This paper presents a generalization of a previous work [ref] where the Loop Quantum Gravity (LQG) quantization of $D=1+2$ gravity with a positive cosmological constant, in the presence of a Barbero Immirzi-like parameter analogous to the one which may be introduced in the four dimensional gravitation theory [ref] (and first introduced in the three-dimensional theory by the authors of [ref] ) was performed using a partial gauge fixing procedure leaving the compact SU(2) group as the residual group of gauge invariance.
Loop Quantization of a Model for D=1+2 (Anti)de Sitter Gravity Coupled to Topological Matter · Introduction · 0.93
Uses its method or tool 1 paper
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We apply to the present model the quantization procedure followed in [ref] .
Loop Quantization of a Model for D=1+2 (Anti)de Sitter Gravity Coupled to Topological Matter · Quantization · 0.98
What it cites · 14 sentences without a confident label
Uses the method or tool of 2 papers
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Specializing to the particular case of a 2-dimensional space manifold with the topology of a cylinder, we have applied a recent LQG quantization scheme of the resulting CS theory [ref] which yields a physical separable Hilbert space with a s-knot basis labeled by spin arrays $j⃗=(j_1,⋯,j_N)$.
The Hilbert space of Chern-Simons theory on the cylinder. A Loop Quantum Gravity approach · Concluding remarks. · 0.93
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This is done with the use of the group averaging method (see [ref] ), based on the Gel'fand triple $S_∘ ⊂ℋ_Gauss ⊂S^'_∘$, being $S_∘^'$ the dual of the spin-networks space $S_∘$.
Background Independent Quantum Gravity: A Status Report · Quantum Theory · 1.00
Compares with 1 paper
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Because of this, $L̂_phys()̱$ is dimensionless, but an operator with dimension of length can be defined as $L̂_phys/√(Λ)$ which has a spectrum very similar to that of the area operator in $D=3+1$ gravity[ With the difference that in $D=3+1$ gravity the area operator is only a partial observable, not even invariant under the space diffeomorphisms [ref] ].
Background Independent Quantum Gravity: A Status Report · Observable · 0.87
Cites as background 5 papers
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In the next section, which is mainly a review of results from [ref] , the basic tools to cast 3d gravity, with (or without) cosmological constant, as a Chern-Simons gauge theory will be presented, and the appearance of the Barbero-Immirzi like parameter will be explained.
A Immirzi-like parameter for 3d quantum gravity · Introduction · 0.94
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Then Barbero pointed out [ref] the possibility of a real realization of canonical variables closely related to Ashtekar's, but the price to be paid is the introduction of an arbitrary real/complex parameter $γ$ known as the Barbero-Immirzi parameter).
Reality Conditions and Ashtekar Variables: a Different Perspective · Introduction · 0.94
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Finally Holst introduced [ref] the most general action integrating all the previous cases, which contains a (on-shell) topological term coupled via the $γ$ parameter.
Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action · Introduction · 0.89
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