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Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity
General Relativity and Quantum Cosmology · Submitted 29 Jun 1996
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric.
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The abstract, sentence by sentence
A Wheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric. The technique introduced here can also be used to produce a couple of other completely well-defined regulated operators including but not exhausting a) the Euclidean Wheeler-DeWitt operator, b)the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, c) length operators, d) Hamiltonian operators of the matter sector and e) the generators of the asymptotic Poincar\'e group including the quantum ADM energy.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A Wheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. | contribution 1.00 | 0.89 | 0.19 | asserted 1.00 |
| s1 | The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric. | result 0.98 | 0.90 | 0.64 | asserted 1.00 |
| s2 | The technique introduced here can also be used to produce a couple of other completely well-defined regulated operators including but not exhausting a) the Euclidean Wheeler-DeWitt operator, b)the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, c) length operators, d) Hamiltonian operators of the matter sector and e) the generators of the asymptotic Poincar\'e group including the quantum ADM energy. | contribution 0.73 | 0.86 | 0.12 | asserted 0.84 |
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Original abstract
A Wheeler-Dewitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. The regulated Wheeler-DeWitt constraint operator is densely defined, does not require any renormalization and the final operator is anomaly-free and at least symmmetric. The technique introduced here can also be used to produce a couple of other completely well-defined regulated operators including but not exhausting a) the Euclidean Wheeler-DeWitt operator, b)the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, c) length operators, d) Hamiltonian operators of the matter sector and e) the generators of the asymptotic Poincar\'e group including the quantum ADM energy.
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References
How the literature uses it
34 arXiv papers cite this in 64 sentences
Papers citing it · 34 sentences without a confident label
Builds on it 1 paper
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As demonstrated in [ref] , the needed method, introduced by Thiemann [ref] in the absence of fermions[In the absence of the torsion tensor, the rapidity in [ref] coincide with the dreiben-contracted extrinsic curvature.], holds also in the presence of the fermionic degrees of freedom.
Relativistic classical theory II. Holonomy-flux representation of gravitational degrees of freedom · Graph representation of the Lorentz connection's rapidity · 0.83
Uses its method or tool 8 papers
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In particular, the regularization procedure presented in this article is based mainly on Thiemann's regularization in canonical loop quantum gravity (CLQG) [ref] .
Relativistic classical theory II. Holonomy-flux representation of gravitational degrees of freedom · Equivalence principle on a lattice · 1.00
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The second, the so-called Lorentzian term $-2γ^2+1/κ∫d^3xN|E|^-1/2K^i_aK^j_bE_[i^aE_j]^b$, can be calculated from the result in [ref] in the standard exact method [ref] .
Implementation of the holonomy representation of the Ashtekar connection in loop quantum gravity · Summary of results · 0.81
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In this subsection, we quantise the Euclidean term and the Lorentzian term in different ways, mimicing the treatment adopted in the minimally coupled modified LQC in references [ref] in which the Lorentzian term is regularised using the Thiemann's trick from the full theory [ref] .
Quantisation ambiguities and the effective dynamics of scalar-tensor theories in loop quantum cosmology · Effective theory of the modified STT-curvature quantisation · 0.88
Compares with it 3 papers
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By expansion of the holonomy around a triangular loop, [equation] one can formulate a significantly different relation than the one in [ref], [equation] One could presume that perhaps this difference was the reason behind formulating the original regularization of LQG [ref] in the way involving the terms up to the order $ε^2$ in expression [ref] but only up to the order $ε$ in formula [ref].
Implementation of the holonomy representation of the Ashtekar connection in loop quantum gravity · Standard implementation · 0.88
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Hence, conversely to the Hamiltonian constraint operator in CLQG [ref] , the quantized Hamiltonian of the lattice gravity, constructed by using the methods presented in this article (see [ref] ), can be considered as a candidate for the Hamiltonian of a quantum gravitational field.
Relativistic classical theory II. Holonomy-flux representation of gravitational degrees of freedom · Piecewise linear graph and a graph with short edges · 0.92
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Here we have a situation very similar to loop quantum gravity [ref] , or parametrized and polymerized scalar field theory [ref] , with a `graph changing' Hamiltonian.
Time evolution as refining, coarse graining and entangling · Massless scalar field in a 2D Lorentzian space time · 0.87
Cites it as background 13 papers
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The first complete construction of the theory, called canonical LQG, was introduced in [ref] and it is the most developed formulation as of today.
Implementation of the holonomy representation of the Ashtekar connection in loop quantum gravity · Introduction · 0.94
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In the seminal work [ref] , Thiemann was able to give a prescription for a well-regulated quantization of the Hamiltonian constraint.
Deriving loop quantum cosmology dynamics from diffeomorphism invariance · Introduction · 0.81
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There graph changing (actually graph refining) Hamiltonian constraints have been defined by Thiemann [ref] .
Time evolution as refining, coarse graining and entangling · Path independence of evolution and consistent embedding maps · 0.86
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