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An algebraic extension of Dirac quantization: Examples
General Relativity and Quantum Cosmology · Submitted 31 May 1994
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints.
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The abstract, sentence by sentence
An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. These issues play an important role especially in the context of non-linear, diffeomorphism invariant theories such as general relativity. Recently, an extension of the required type was proposed by one of us using algebraic quantization methods. In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. However, prior knowledge of general relativity is not assumed in the main discussion. Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. | background 0.52 | 0.09 | 0.09 | asserted 1.00 |
| s1 | These issues play an important role especially in the context of non-linear, diffeomorphism invariant theories such as general relativity. | background 0.98 | 0.07 | 0.03 | asserted 0.96 |
| s2 | Recently, an extension of the required type was proposed by one of us using algebraic quantization methods. | background 0.69 | 0.63 | 0.05 | asserted 1.00 |
| s3 | In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. | contribution 0.75 | 0.89 | 0.09 | asserted 1.00 |
| s4 | The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. | background 0.58 | 0.50 | 0.04 | asserted 0.93 |
| s5 | However, prior knowledge of general relativity is not assumed in the main discussion. | background 0.75 | 0.35 | 0.03 | asserted 1.00 |
| s6 | Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. | interpretation 0.77 | 0.72 | 0.37 | asserted 1.00 |
| s7 | In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems. | result 0.62 | 0.77 | 0.25 | asserted 0.99 |
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The paper
Original abstract
An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. These issues play an important role especially in the context of non-linear, diffeomorphism invariant theories such as general relativity. Recently, an extension of the required type was proposed by one of us using algebraic quantization methods. In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. However, prior knowledge of general relativity is not assumed in the main discussion. Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems.
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References
How the literature uses it
7 arXiv papers cite this in 9 sentences
Papers citing it · 7 sentences without a confident label
Uses its method or tool 2 papers
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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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We will first start with a simple system, where all of the the steps of the algebraic extension to Dirac's quantization procedure, outlined in [ref] can be carried out to completion.
Instanton representanton of Plebanski gravity. Gravitational coherent states · Quantum harmonic oscillator formalism · 0.80
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