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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
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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.
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- 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.
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- 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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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- 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.
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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.
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- 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.
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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
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Discreteness of area and volume in quantum gravity
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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
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Real Ashtekar variables for Lorentzian signature space-times
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Black Hole Entropy from Loop Quantum Gravity
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Holst, S. (1996) · Physical Review D · doi:10.1103/physrevd.53.5966 · Cited by 576
Knot Theory and Quantum Gravity
Rovelli, C., & Smolin, L. (1988) · Physical Review Letters · doi:10.1103/physrevlett.61.1155 · Cited by 463
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Quantum gravity and Regge calculus
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1 Quantum Gravity and Regge CalculusPreprint
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Immirzi parameter in quantum general relativity
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Volume Operator in Discretized Quantum Gravity
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Reality conditions and Ashtekar variables: A different perspective
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Spectrum of the volume operator in quantum gravity
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Stochastic path integrals and open quantum systems
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From Euclidean to Lorentzian general relativity: The real way
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Real-polynomial formulation of general relativity in terms of connections
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Is Barbero's Hamiltonian formulation a Gauge Theory of Lorentzian Gravity?
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Non-perturbative solutions for lattice quantum gravity
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Path integral for the Hilbert-Palatini and Ashtekar gravity
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Self-citationProduction or annihilation of positrons with bound electrons
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Cited by
Sorted by citations
The Spin-Foam Approach to Quantum Gravity
Perez, A. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-3 · Cited by 677
Quantum gravity: a progress report
Carlip, S. (2001) · Reports on Progress in Physics · doi:10.1088/0034-4885/64/8/301 · Cited by 449
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Pérez, A. (2017) · Reports on Progress in Physics · doi:10.1088/1361-6633/aa7e14 · Cited by 221
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Livine, E. R., & Simone Speziale, D. (2008) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/81/50004 · Cited by 177
SU(2)loop quantum gravity seen from covariant theory
Alexandrov, S., & Livine, E. R. (2003) · Physical Review D · doi:10.1103/physrevd.67.044009 · Cited by 141
Self-citation
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
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