On the generality of refined algebraic quantization
Open accessAbstract
The Dirac quantization `procedure' for constrained systems is well known to have many subtleties and ambiguities. Within this ill-defined framework, we explore the generality of a particular interpretation of the Dirac procedure known as refined algebraic quantization. We find technical conditions under which refined algebraic quantization can reproduce the general implementation of the Dirac scheme for systems whose constraints form a Lie algebra with structure constants. The main result is that, under appropriate conditions, the choice of an inner product on the physical states is equivalent to the choice of a `rigging map' in refined algebraic quantization.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The inner product on the space of solutions of the Wheeler-DeWitt equation must be gauge-fixed, and little is understood about the resulting Hilbert space.
“Further complications come from the fact that the inner product on the space of solutions of the Wheeler-DeWitt equation must be gauge-fixed [282, 144]; again, little is understood about the resulting Hilbert space.”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [144]
The Introduction and §2.1 describe difficulties and ambiguities in defining an inner product and physical Hilbert space for solutions of quantum constraints. But §2.2 defines the inner product through a rigging map, not as something that must be gauge-fixed. The paper discusses constrained systems generally and does not establish the claim specifically for Wheeler–DeWitt solutions. (LLM, conf=0.85)
Experimental — accuracy not yet measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
Quantum observables and recollapsing dynamics
Marolf, D. (1995) · Classical and Quantum Gravity · doi:10.1088/0264-9381/12/5/011 · Cited by 153 · see citing works
Self-citationProposal for solving the ‘‘problem of time’’ in canonical quantum gravity
Wald, R. M. (1993) · Physical Review D · doi:10.1103/physrevd.48.r2377 · Cited by 59 · see citing works
Flagged first, then by citations
Research works printed but not linked: 23
- Tyutin I V 1975 Lebedev preprint FIAN • 39
- Henneaux M and Teitelboim C 1992 Quantization of Gauge Systems (Princeton, NJ: Princeton University Press)
- Woodhouse N 1980 Geometric Quantization (Oxford: Oxford University Press)
- Klauder J 1999 Product representations and the quantization of constrained systems Fadeev Festschrift to be published (Klauder J 1998 Preprint quant-ph/9811051) · arXiv
- Klauder J 1999 Coherent state path integrals at (nearly) 40 Proc Florence Path Integral Conf. to be published (Klauder J 1998 Preprint quant-ph/9810043) · arXiv
- Grundling H and Lledó 1998 Local quantum constraints Preprint math-ph/9812022 · arXiv
- Ashtekar A 1991 Lectures on Non-Perturbative Canonical Gravity (River Edge, NJ: World Scientific)
- Ashtekar A and Tate R 1994 Preprint gr-qc/9405073) · arXiv
- Mena Marugán G 1996 J. Math. Phys. 37 196-205 (Mena Marugán G 1995 Preprint gr-qc/9506038) · arXiv
- Wald R 1993 Phys. Rev. D 48 R2377 Higuchi A and Wald R 1995 Phys. Rev. D 51 544 (Higuchi A and Wald R 1994 Preprint gr-qc/9407038) · arXiv
- Ashtekar A, Lewandowski J, Marolf D, Mourão J and Thiemann T 1995 Preprint gr-qc/9504018) · arXiv
- Marolf D 1997 Symplectic Singularities and Geometry of Gauge Fields (Warsaw: Banach) (Marolf D 1995 Preprint gr-qc/9508015) · arXiv
- Landsman N 1995 J. Geom. Phys. 15 285-319 (Landsman N 1993 Preprint hep-th/9305088) · arXiv
- Marolf D 1995 Class. Quantum Grav. 12 1199 (Marolf D 1994 Preprint gr-qc/9404053) · arXiv
- Marolf D 1995 Class. Quantum Grav. 12 1441 (Marolf D 1994 Preprint gr-qc/9409049) · arXiv
- Marolf D 1995 Class. Quantum Grav. 12 2469-86 (Marolf D 1994 Preprint gr-qc/9412016) · arXiv
- Marolf D 1996 Phys. Rev. D 53 6979-90 (Marolf D 1996 Preprint gr-qc/9602019) · arXiv
- Duval C, Elhadad J, Gotay M J and Tuynman G M 1990 Proc. CRM Workshop on Hamiltonian Systems, Transformation Groups and Spectral Transform Methods ed J Harnad and J E Marsden (Montreal: CRM)
- DeWitt B 1962 The quantization of geometry Gravitation: an Introduction to Current Research ed L Witten (New York: Wiley)
- Kuchař K 1992 Time and interpretations of quantum gravity Proc. 4th Canadian Conf. on General Relativity and Relativistic Astrophysics ed G Kunstatter, D E Vincent and J G Williams (River Edge, NJ: World Scientific)
- Isham C J 1997 General Relativity and Gravitation (Florence, 1995) ed M Francaviglia et al (River Edge, NJ: World Scientific) pp 167-209 (
- Isham C J 1995 Preprint gr-qc/9510063) · arXiv
- Bohm A 1986 Quantum Mechanics: Foundations and Applications (New York: Springer) p 100
Cited by
Sorted by citations
Background independent quantum gravity: a status report
Ashtekar, A., & Lewandowski, J. (2004) · Classical and Quantum Gravity · doi:10.1088/0264-9381/21/15/r01 · Cited by 1,951 · see citing works
Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
- Authors
- Giulini, D., & Marolf, D.
- Affiliations
- Institute for Theoretical Physics (CH)
- Syracuse University (US)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1999-01-01
- Volume
- 16
- Issue
- 7
- Pages
- 2479-2488
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2073992954
- Cited by
- 144
- References
- 2
- Citations by year
- 2026: 2 · 2025: 12 · 2024: 8 · 2023: 19 · 2022: 2 · 2021: 3 · 2020: 2 · 2018: 2 · 2017: 2 · 2016: 7 · 2015: 3 · 2014: 3 · 2013: 4 · 2012: 4
- FWCI
- 3.58
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9722362)
- Syracuse University
- Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Keywords
- Dirac quantization, constrained systems, Dirac brackets, structure constants, canonical quantization
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
