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QSD IV : 2+1 Euclidean Quantum Gravity as a model to test 3+1 Lorentzian Quantum Gravity
General Relativity and Quantum Cosmology · Submitted 10 May 1997
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Thus, by suitable restriction of the solution space, we can recover all former results which gives confidence in the new quantization methods.
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The abstract, sentence by sentence
The quantization of Lorentzian or Euclidean 2+1 gravity by canonical methods is a well-studied problem. However, the constraints of 2+1 gravity are those of a topological field theory and therefore resemble very little those of the corresponding Lorentzian 3+1 constraints. In this paper we canonically quantize Euclidean 2+1 gravity for arbitrary genus of the spacelike hypersurface with new, classically equivalent constraints that maximally probe the Lorentzian 3+1 situation. We choose the signature to be Euclidean because this implies that the gauge group is, as in the 3+1 case, SU(2) rather than SU(1,1). We employ, and carry out to full completion, the new quantization method introduced in preceding papers of this series which resulted in a finite 3+1 Lorentzian quantum field theory for gravity. The space of solutions to all constraints turns out to be much larger than the one as obtained by traditional approaches, however, it is fully included. Thus, by suitable restriction of the solution space, we can recover all former results which gives confidence in the new quantization methods. The meaning of the remaining "spurious solutions" is discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The quantization of Lorentzian or Euclidean 2+1 gravity by canonical methods is a well-studied problem. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | However, the constraints of 2+1 gravity are those of a topological field theory and therefore resemble very little those of the corresponding Lorentzian 3+1 constraints. | gap 0.64 | 0.08 | 0.03 | asserted 1.00 |
| s2 | In this paper we canonically quantize Euclidean 2+1 gravity for arbitrary genus of the spacelike hypersurface with new, classically equivalent constraints that maximally probe the Lorentzian 3+1 situation. | contribution 1.00 | 0.95 | 0.26 | asserted 1.00 |
| s3 | We choose the signature to be Euclidean because this implies that the gauge group is, as in the 3+1 case, SU(2) rather than SU(1,1). | method 0.97 | 0.58 | 0.03 | asserted 1.00 |
| s4 | We employ, and carry out to full completion, the new quantization method introduced in preceding papers of this series which resulted in a finite 3+1 Lorentzian quantum field theory for gravity. | method 0.91 | 0.88 | 0.07 | asserted 1.00 |
| s5 | The space of solutions to all constraints turns out to be much larger than the one as obtained by traditional approaches, however, it is fully included. | result 1.00 | 0.93 | 0.19 | asserted 1.00 |
| s6 | Thus, by suitable restriction of the solution space, we can recover all former results which gives confidence in the new quantization methods. | interpretation 0.87 | 0.78 | 0.24 | asserted 0.80 |
| s7 | The meaning of the remaining "spurious solutions" is discussed. | interpretation 0.73 | 0.72 | 0.06 | unclear 0.83 |
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Original abstract
The quantization of Lorentzian or Euclidean 2+1 gravity by canonical methods is a well-studied problem. However, the constraints of 2+1 gravity are those of a topological field theory and therefore resemble very little those of the corresponding Lorentzian 3+1 constraints. In this paper we canonically quantize Euclidean 2+1 gravity for arbitrary genus of the spacelike hypersurface with new, classically equivalent constraints that maximally probe the Lorentzian 3+1 situation. We choose the signature to be Euclidean because this implies that the gauge group is, as in the 3+1 case, SU(2) rather than SU(1,1). We employ, and carry out to full completion, the new quantization method introduced in preceding papers of this series which resulted in a finite 3+1 Lorentzian quantum field theory for gravity. The space of solutions to all constraints turns out to be much larger than the one as obtained by traditional approaches, however, it is fully included. Thus, by suitable restriction of the solution space, we can recover all former results which gives confidence in the new quantization methods. The meaning of the remaining "spurious solutions" is discussed.
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References
How the literature uses it
16 arXiv papers cite this in 74 sentences
Papers citing it · 38 sentences without a confident label
Builds on it 2 papers
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It follows that the problem reduces to the $2+1$-dimensional one, which has been sketched in [ref] , just that here we consider the special case of a spherical topology.
On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity · Imposition of reduction constraints · 0.96
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In [ref] they found that the 6j (a physical state in 3d) is annihilated by a suitable quantization of the 3d scalar constraint $F=0$ rewritten, following [ref] , as $EEF$.
Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint · Conclusions · 0.84
Uses its method or tool 4 papers
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We will take fist steps in investigating this problem in the present paper, using as an embedding of reduced theory quantum states into the reduced sector of the full theory the proposal of [ref] , which uses crucial properties of the $2+1$-dimensional volume operator [ref] .
On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity · Introduction · 0.92
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Using the formula Fab of curvature together with Thiemann's trick in 2+1 dimensional LQG [ref] ϵ^ijkϵ_abE^a_jE^b_k=12(8πGγ)^2ϵ_ijkϵ^ab{A_a^j,V}{A_b^k,V},the operator of the gravitational part of the Hamiltonian constraint _=-116πGγ^2∫_E^a_iE^b_j√(q)ϵ^ij_kF_ab^k can be given by [equation] where $h_K$ is short for $h^(_K)_K$, the holonomy along edge $e_K$ in the diagonal direction $K$.
2+1 dimensional loop quantum cosmology of Bianchi I models · Construction of Hamiltonian operator · 1.00
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In this section, we investigate the solution of the quantum Hamilton constraint constructed using the Thiemann strategy [ref] .
Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit · Investigating the Thiemann regularised quantum Hamilton constraint of 3d LQG · 0.97
Uses its data or values 1 paper
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According to [ref] , the eigenvalues of $W$ on a 3-valent gauge-invariant node are given by (see equation A.6 in [ref] ): W_3(j_1,j_2,j_3)=1/4√(9/2(Δ_j_1Δ_j_2+Δ_j_1 Δ_j_3+Δ_j_2 Δ_j_3)-9/4(Δ_j_1^2+Δ_j_2^2+Δ_j_3 ^2)-1/2(Δ_j_1+Δ_j_2+Δ_j_3 )) .
On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity · Properties of $^r$ and action on a kink · 0.90
Compares with it 2 papers
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If we instead consider the first ordering of $$, which is closer to [ref] , then [ref] and [ref] also show different scaling behaviours in the regime of large quantum numbers.
On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity · Comparison of the classically and quantum reduced algebras · 0.99
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As expected, this operator is identical to the one of [ref] up to the factor of $Δl_i$.
A quantum reduction to spherical symmetry in loop quantum gravity · Introduction · 0.98
Cites it as background 4 papers
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In the full theory of 2+1 dimensional LQG [ref] , the phase space is spanned by a canonical pair ($A_a^i, E^a_i$), with $A_a^i$ an $(2)$ connection and $E^a_i=δ_ijϵ^abe_b^i$ the momentum conjugate to $A_a^i$.
2+1 dimensional loop quantum cosmology of Bianchi I models · The canonical pair of 2+1 dimensional Bianchi I models · 0.84
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In particular, the quantization programs of loop quantum gravity and spin foams have been successfully applied, on the kinematical aspects of quantum geometry [ref] as well as on its dynamics [ref] .
The Hamiltonian constraint in 3d Riemannian loop quantum gravity · Introduction · 0.86
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In particular, it was one of the first example of the loop quantization programme [ref] and has since been completed (see e.g. [ref] , see also [ref] for quantum groups and/or Chern-Simons approaches).
A Immirzi-like parameter for 3d quantum gravity · Quantization of lengths with the Immirzi parameter · 0.97
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