The DOI matches its Crossref record; 18 of the 36 references it cites were found in a registry or an index, and 18 are not checked yet; none of the 12 sources its claims cite is checked for support yet.
SO (4, C )-covariant Ashtekar-Barbero gravity and the Immirzi parameter
Open accessAbstract
An so (4, C )-covariant Hamiltonian formulation of a family of generalized Hilbert-Palatini actions depending on a parameter (the so-called Immirzi parameter) is developed. It encompasses the Ashtekar-Barbero gravity which serves as a basis of quantum loop gravity. Dirac quantization of this system is constructed. Next we study the dependence of the quantum system on the Immirzi parameter. The path-integral quantization shows no dependence on it. A method of modifying the loop approach in accordance with the formalism developed here is briefly outlined.
Checked 2026-10-06 · 18 references not checked yet
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
23 No basis given · 6 Not checked yet · 1 Own interpretation · 1 Own result
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The path-integral quantization shows no dependence on it.”
- Also stated in: § 4.1. Path-integral quantization · hedged
“Thus we arrive at the conclusion that at least at the formal level, the path integral for quantum gravity is independent of the Immirzi parameter.”
- Also stated in: § 5. Conclusions · boosted
“It has been shown within the framework of the formal path-integral quantization that the formulations with different values of β should all be equivalent.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > Abstract
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: The formal path-integral manipulations are sufficient to establish the β independence of the quantum theory.
- Also stated in: § 4.1. Path-integral quantization · hedged
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“An so(4, C)-covariant Hamiltonian formulation of a family of generalized Hilbert-Palatini actions depending on a parameter (the so-called Immirzi parameter) is developed.”
- Also stated in: § 5. Conclusions · hedged
“In this paper we have suggested a new Hamiltonian formulation of general relativity based on the full SO(3, 1) gauge group.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § 5. Conclusions · hedged
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Dirac quantization of this system is constructed.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > Abstract
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note, that in this basis the Hamiltonian constraint can be expressed through the dynamical multiplet only,”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > Appendix B. Dual representation
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As a result the dependence of the path integral on β is contained in three multipliers ( 35)-(37) which cancel each other.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 4.1. Path-integral quantization
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Since {G X , g ij } D = 0, the complex gauge transformations do not destroy the reality of the metric.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 3. Covariant canonical formulation
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Moreover, it is covariant under arbitrary transformations of the basis of the adjoint representation.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 2. Generalized Hilbert-Palatini action
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Thus the Wilson loop operator can be constructed as”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 4.2. Loop approach
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Then ˜ α are first-class constraints with the algebra presented in appendix C.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 3. Covariant canonical formulation
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“They obey the following algebra:”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > Appendix C. Constraint algebra
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- It may state two separate claims as one. 4
- The quote only partly states this claim. 1
- Probably not a substantial claim. 2
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In the covariant formalism we have a unique connection for all values of β instead of a oneparameter family of connections in the Ashtekar-Barbero gravity.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 4.2. Loop approach
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Let us note that in the Barbero approach based on the SU (2) gauge subgroup the Wick transformation changes the connection rather than the triad.In our case the connection is unique but there are two triad multiplets.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 4.2. Loop approach
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Thus we arrive at the conclusion that at least at the formal level, the path integral for quantum gravity is independent of the Immirzi parameter.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 4.1. Path-integral quantization
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
- The authors state this as an assumption: The formal path-integral manipulations are sufficient to establish the β independence of the quantum theory.
None of the sources cited for this claim have been checked yet.
In the paper
“In this framework a set of remarkable physical results has been obtained, such as the discrete spectra of the area and volume operators [4,5] and a derivation of the Bekenstein-Hawking formula for the black hole entropy [6].”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 1. Introduction
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Of course, the observed duality is trivial in the sense that it is only a change of variables.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > Appendix B. Dual representation
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“In [12] Holst has shown that in the 'time' gauge this action reproduces Barbero's formulation [8] and the parameter β plays the role of the Immirzi parameter.”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 2. Generalized Hilbert-Palatini action
What the sources say
- Not checked yetcites [12] Frittelli S, 1996, Class. Quantum Grav. source
- Not checked yetcites [8] Rovelli C, 1997
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
The authors state this as an assumption.
The formal path-integral manipulations are sufficient to establish the β independence of the quantum theory.
“Thus we arrive at the conclusion that at least at the formal level, the path integral for quantum gravity is independent of the Immirzi parameter.”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > 2. Generalized Hilbert-Palatini action
“In [12] Holst has shown that in the 'time' gauge this action reproduces Barbero's formulation [8]”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > References
“12. Loll R 1995 Phys. Rev. Lett. 75 3048”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > Appendix C. Constraint algebra
“† The idea that the problem of the Immirzi parameter requires a group larger than SU (2) has been stressed by Immirzi himself in the work where the problem was discovered[9].”
§ SO(4, C)-covariant Ashtekar-Barbero gravity and the Immirzi parameter > References
“9. Lewandowski J 1995 The operators of quantum gravity Lecture at the Workshop on Canonical and Quantum Gravity (Warsaw)”
Citations
Cites
New Variables for Classical and Quantum Gravity
Ashtekar, A. (1986) · Physical Review Letters · doi:10.1103/physrevlett.57.2244 · Cited by 1,663
Discreteness of area and volume in quantum gravity
Rovelli, C., & Smolin, L. (1995) · Nuclear Physics B · doi:10.1016/0550-3213(95)00150-q · Cited by 1,365
New Hamiltonian formulation of general relativity
Ashtekar, A. (1987) · Physical Review D · doi:10.1103/physrevd.36.1587 · Cited by 1,119
Quantum geometry and black hole entropy
Ashtekar, A., Baez, J., Corichi, A., & Krasnov, K. (1998) · Physical Review Letters · doi:10.1103/physrevlett.80.904 · Cited by 1,105
Lectures on Non-Perturbative Canonical Gravity
Ashtekar, A. (1991) · Advanced Series in Astrophysics and Cosmology · doi:10.1142/1321 · Cited by 872
Quantum theory of geometry: I. Area operators
Ashtekar, A., & Lewandowski, J. (1997) · Classical and Quantum Gravity · doi:10.1088/0264-9381/14/1a/006 · Cited by 852
Loop space representation of quantum general relativity
Rovelli, C., & Smolin, L. (1990) · Nuclear Physics B · doi:10.1016/0550-3213(90)90019-a · Cited by 806
Real Ashtekar variables for Lorentzian signature space-times
Barbero G., J. F. (1995) · Physical Review D · doi:10.1103/physrevd.51.5507 · Cited by 758
Black Hole Entropy from Loop Quantum Gravity
Rovelli, C. (1996) · Physical Review Letters · doi:10.1103/physrevlett.77.3288 · Cited by 660
Baym, G. (2018) · CRC Press eBooks · doi:10.1201/9780429499265 · Cited by 580
Research works printed but not linked: 2
- Rovelli C, 1997, Living Rev.
- Faddeev L D, 1969, Teor. Mat. Fiz., 1, 3
Cited by
Sorted by citations
Items 101–112 of 112
Gravity, Geometry, and the Quantum
Sahlmann, H. (2014) · Springer handbooks · doi:10.1007/978-3-642-41992-8_37 · Cited by 1
THE ROLE OF THE TIME GAUGE IN THE SECOND-ORDER FORMALISM
Cianfrani, F., & Montani, G. (2008) · International Journal of Modern Physics A · doi:10.1142/s0217751x08040081 · Cited by 1
Superspace worldline formalism approach to quantum gravity: dimensional reduction and holography
Roux, J.-B. (2024) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ad56ef · Cited by 1
Hamiltonian gravity in tetrad-connection variables
Duque, E. I. (2025) · Physical review. D/Physical review. D. · doi:10.1103/3r98-k11x · Cited by 2
Лоренц-ковариантная петлевая квантовая гравитация
Alexandrov, S. Y. (2004) · Теоретическая и математическая физика · doi:10.4213/tmf62
Self-citationSO(2,1) Connection in Timelike 3+1 FoliationPreprint
Perlov, L. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1903.01662
Hamiltonian and Diffeomorphism Constraints Generalized for Timelike and Spacelike 3+1 FoliationPreprint
Perlov, L. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2001.03773
Gravity, Geometry, and the Quantum
Sahlmann, H. (2014) · Springer eBooks · doi:10.1007/978-3-662-46035-1_37
Bonzom, V. (2010) · theses.fr (ABES) · doi:10.70675/248ccc62z7e63z4f22z89ddz3e703205a026
Aspects de gravitation quantique
Liu, H. (2019) · theses.fr (ABES) · doi:10.70675/914be533zcf19z4b75z8627z3d22268906b7
Metadata
- Authors
- Alexandrov, S.
- Affiliations
- St Petersburg University (RU)
- Corresponding author
- Alexandrov, S
- ORCID
- Alexandrov, S
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2000-09-26
- Volume
- 17
- Issue
- 20
- Pages
- 4255-4268
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2011688977
- Semantic Scholar
- 88ba3006edfc3fce317b83940ccc2b0c723ef608
- Cited by
- 112
- References
- 36
- Citations by year
- 2025: 1 · 2024: 1 · 2023: 1 · 2022: 1 · 2021: 3 · 2020: 4 · 2019: 5 · 2018: 3 · 2017: 4 · 2016: 1 · 2015: 4 · 2014: 4 · 2013: 8 · 2012: 10
- FWCI
- 3.84
- 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)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- Immirzi parameter, Hamiltonian formulation, Dirac quantization, loop quantum gravity, path integral 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.
A language model drew the claims from the paper on 2026-10-06, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-06 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
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
