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Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models
General Relativity and Quantum Cosmology · Submitted 29 Nov 2004
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The physical Hilbert space can then be be obtained after subtracting this normal ordering correction.
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The abstract, sentence by sentence
This is the third paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper: These are systems with an $SL(2,\Rl)$ gauge symmetry and the complications arise because non -- compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the Master Constraint does not contain the point zero. However, the minimum of the spectrum is of order $\hbar^2$ which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to $\hbar$ normal ordering constants). The physical Hilbert space can then be be obtained after subtracting this normal ordering correction.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This is the third paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. | background 0.97 | 0.53 | 0.03 | asserted 1.00 |
| s1 | In this work we analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper: These are systems with an $SL(2,\Rl)$ gauge symmetry and the complications arise because non -- compact semisimple Lie groups are not amenable (have no finite translation invariant measure). | contribution 0.93 | 0.74 | 0.05 | asserted 1.00 |
| s2 | This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the Master Constraint does not contain the point zero. | result 0.87 | 0.83 | 0.16 | asserted 0.84 |
| s3 | However, the minimum of the spectrum is of order $\hbar^2$ which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to $\hbar$ normal ordering constants). | interpretation 0.54 | 0.86 | 0.24 | asserted 0.73 |
| s4 | The physical Hilbert space can then be be obtained after subtracting this normal ordering correction. | result 0.44 | 0.77 | 0.22 | asserted 0.94 |
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Original abstract
This is the third paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper: These are systems with an $SL(2,\Rl)$ gauge symmetry and the complications arise because non -- compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the Master Constraint does not contain the point zero. However, the minimum of the spectrum is of order $\hbar^2$ which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to $\hbar$ normal ordering constants). The physical Hilbert space can then be be obtained after subtracting this normal ordering correction.
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4 arXiv papers cite this in 4 sentences
Papers citing it · 3 sentences without a confident label
Uses its method or tool 1 paper
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The method combines refined algebraic [ref] and master constraint [ref] quantization and results in a physical Hilbert space that can be expressed as a $L^2$ Hilbert space over a continuous (momentum) parameter $ρ$ [ref] .
Can chaos be observed in quantum gravity? · Quantization (Refined Topology) · 0.89
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