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Dynamics in canonical models of loop quantum gravity
General Relativity and Quantum Cosmology · Submitted 1 Oct 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Most importantly, the new Hamiltonian can be constructed as a symmetric operator, and is therefore a mathematically consistent candidate for a generator of physical time evolution in deparametrized models.
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The abstract, sentence by sentence
In this thesis we consider the problem of dynamics in canonical loop quantum gravity, primarily in the context of deparametrized models, in which a scalar field is taken as a physical time variable for the dynamics of the gravitational field. The dynamics of the quantum states of the gravitational field is then generated directly by a physical Hamiltonian operator, instead of being implicitly defined through the kernel of a Hamiltonian constraint. We introduce a new construction of a Hamiltonian operator for loop quantum gravity, which has both mathematical and practical advantages in comparison to earlier proposals. Most importantly, the new Hamiltonian can be constructed as a symmetric operator, and is therefore a mathematically consistent candidate for a generator of physical time evolution in deparametrized models. We develop methods for approximately evaluating the dynamics generated by a given physical Hamiltonian, even if an exact solution to the eigenvalue problem of the Hamiltonian cannot be achieved. We also introduce a new representation for intertwiners in loop quantum gravity, based on projecting intertwiners onto coherent states of angular momentum, and in which intertwiners are represented as polynomials of certain complex variables, and operators in loop quantum gravity are expressed as differential operators acting on these variables. In addition to reviewing the results of the author's scientific work, this thesis also gives a thorough introduction to the basic framework of canonical loop quantum gravity, and a self-contained presentation of the graphical formalism for SU(2) recoupling theory, which is the invaluable tool for performing practical calculations in loop quantum gravity. The author therefore hopes that parts of this thesis could serve as a comprehensible source of information for anyone interested in learning the elements of loop quantum gravity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this thesis we consider the problem of dynamics in canonical loop quantum gravity, primarily in the context of deparametrized models, in which a scalar field is taken as a physical time variable for the dynamics of the gravitational field. | background 0.83 | 0.63 | 0.05 | asserted 0.92 |
| s1 | The dynamics of the quantum states of the gravitational field is then generated directly by a physical Hamiltonian operator, instead of being implicitly defined through the kernel of a Hamiltonian constraint. | background 0.72 | 0.25 | 0.11 | asserted 1.00 |
| s2 | We introduce a new construction of a Hamiltonian operator for loop quantum gravity, which has both mathematical and practical advantages in comparison to earlier proposals. | contribution 1.00 | 0.94 | 0.21 | asserted 1.00 |
| s3 | Most importantly, the new Hamiltonian can be constructed as a symmetric operator, and is therefore a mathematically consistent candidate for a generator of physical time evolution in deparametrized models. | result 0.47 | 0.89 | 0.31 | asserted 0.71 |
| s4 | We develop methods for approximately evaluating the dynamics generated by a given physical Hamiltonian, even if an exact solution to the eigenvalue problem of the Hamiltonian cannot be achieved. | method 0.64 | 0.94 | 0.07 | asserted 0.95 |
| s5 | We also introduce a new representation for intertwiners in loop quantum gravity, based on projecting intertwiners onto coherent states of angular momentum, and in which intertwiners are represented as polynomials of certain complex variables, and operators in loop quantum gravity are expressed as differential operators acting on these variables. | contribution 0.88 | 0.94 | 0.07 | asserted 1.00 |
| s6 | In addition to reviewing the results of the author's scientific work, this thesis also gives a thorough introduction to the basic framework of canonical loop quantum gravity, and a self-contained presentation of the graphical formalism for SU(2) recoupling theory, which is the invaluable tool for performing practical calculations in loop quantum gravity. | contribution 0.62 | 0.46 | 0.04 | asserted 0.99 |
| s7 | The author therefore hopes that parts of this thesis could serve as a comprehensible source of information for anyone interested in learning the elements of loop quantum gravity. | future work 0.67 | 0.08 | 0.03 | hedged possibility 1.00 |
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The paper
Original abstract
In this thesis we consider the problem of dynamics in canonical loop quantum gravity, primarily in the context of deparametrized models, in which a scalar field is taken as a physical time variable for the dynamics of the gravitational field. The dynamics of the quantum states of the gravitational field is then generated directly by a physical Hamiltonian operator, instead of being implicitly defined through the kernel of a Hamiltonian constraint. We introduce a new construction of a Hamiltonian operator for loop quantum gravity, which has both mathematical and practical advantages in comparison to earlier proposals. Most importantly, the new Hamiltonian can be constructed as a symmetric operator, and is therefore a mathematically consistent candidate for a generator of physical time evolution in deparametrized models. We develop methods for approximately evaluating the dynamics generated by a given physical Hamiltonian, even if an exact solution to the eigenvalue problem of the Hamiltonian cannot be achieved. We also introduce a new representation for intertwiners in loop quantum gravity, based on projecting intertwiners onto coherent states of angular momentum, and in which intertwiners are represented as polynomials of certain complex variables, and operators in loop quantum gravity are expressed as differential operators acting on these variables. In addition to reviewing the results of the author's scientific work, this thesis also gives a thorough introduction to the basic framework of canonical loop quantum gravity, and a self-contained presentation of the graphical formalism for SU(2) recoupling theory, which is the invaluable tool for performing practical calculations in loop quantum gravity. The author therefore hopes that parts of this thesis could serve as a comprehensible source of information for anyone interested in learning the elements of loop quantum gravity.
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How the literature uses it
3 arXiv papers cite this in 6 sentences
Papers citing it · 2 sentences without a confident label
Cites it as background 2 papers
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For a more detailed presentation of the foundations of loop quantum gravity, we refer the reader to [ref] .
Scalar curvature operator for models of loop quantum gravity on a cubical graph · Loop quantum gravity · 1.00
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(See [ref] or [ref] .)
Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity · Exercises · 0.86
What it cites · 119 sentences without a confident label
Uses the method or tool of 9 papers
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The general idea for constructing the space $H_vtx$ is to start with the space $H_G^Γ$ of gauge-invariant states[Here we are following the construction presented in [ref] and [ref] , rather than the original construction of [ref] .
Hamiltonian operator for loop quantum gravity coupled to a scalar field · The vertex Hilbert space · 0.89
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The general idea for constructing the space $H_vtx$ is to start with the space $H_G^Γ$ of gauge-invariant states[Here we are following the construction presented in [ref] and [ref] , rather than the original construction of [ref] .
A symmetric scalar constraint for loop quantum gravity · The vertex Hilbert space · 0.89
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The general idea for constructing the space $H_vtx$ is to start with the space $H_G^Γ$ of gauge-invariant states[Here we are following the construction presented in [ref] and [ref] , rather than the original construction of [ref] .
New scalar constraint operator for loop quantum gravity · The vertex Hilbert space · 0.89
Uses data or values from 1 paper
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Some sample results of expectation values calculated from A(t)-pert are shown in VS2B–[ref] for the scalar field model, and in VD2B–[ref] for the dust model[The plots displayed in this chapter are used here with the permission of Mehdi Assanioussi, who created them for the article [ref] on the basis of our joint calculations.].
Time evolution in deparametrized models of loop quantum gravity · Perturbation theory · 0.92
Compares with 1 paper
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The first version differs from the operator defined in [ref] essentially only by the way in which we choose to symmetrize the curvature operator, while in the second version the quantization of the Euclidean part of the constraint is also modified in comparison to [ref] .
New scalar constraint operator for loop quantum gravity · A new Hamiltonian constraint operator · 0.90
Cites as support 2 papers
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On the other hand, the continuous ”moduli” do not seem to have any physical relevance, in that not any known operator in loop quantum gravity is sensitive to their values (as pointed out by Thiemann in [ref] and [ref] ).
A length operator for canonical quantum gravity · Diffeomorphism invariance · 0.98
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On the other hand, the continuous ”moduli” do not seem to have any physical relevance, in that not any known operator in loop quantum gravity is sensitive to their values (as pointed out by Thiemann in [ref] and [ref] ).
QSD III : Quantum Constraint Algebra and Physical Scalar Product in Quantum General Relativity · Diffeomorphism invariance · 0.98
Cites as background 48 papers
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The two models which will be considered are the free scalar field model proposed by Rovelli and Smolin [ref] (and developed further in [ref] and [ref] ), and the non-rotational dust model, which was introduced classically by Brown and Kuchař [ref] , and first studied in the context of loop quantum gravity by Husain and Pawłowski [ref] .
Gravity quantized · Classical theory of deparametrized models · 0.93
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The two models which will be considered are the free scalar field model proposed by Rovelli and Smolin [ref] (and developed further in [ref] and [ref] ), and the non-rotational dust model, which was introduced classically by Brown and Kuchař [ref] , and first studied in the context of loop quantum gravity by Husain and Pawłowski [ref] .
Hamiltonian operator for loop quantum gravity coupled to a scalar field · Classical theory of deparametrized models · 0.93
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The two models which will be considered are the free scalar field model proposed by Rovelli and Smolin [ref] (and developed further in [ref] and [ref] ), and the non-rotational dust model, which was introduced classically by Brown and Kuchař [ref] , and first studied in the context of loop quantum gravity by Husain and Pawłowski [ref] .
The physical hamiltonian in nonperturbative quantum gravity · Classical theory of deparametrized models · 0.93
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