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.
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- Findingreasoning
Path-integral quantization of the theory shows no dependence on the Immirzi parameter.
“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.”
Quote in the paperSubstantial claim- 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
- Findingown data
A Hamiltonian formulation of generalized Hilbert-Palatini actions depending on the Immirzi parameter has been constructed that is covariant under so(4,C).
“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.”
Quote in the paperSubstantial claim - Also stated in: § 5. Conclusions · hedged
- Findingnone given
Dirac quantization of the system has been constructed.
“Dirac quantization of this system is constructed.”
Quote in the paperSubstantial claim - Findingnone given
In the dual basis the Hamiltonian constraint can be expressed through the dynamical multiplet alone.
“Note, that in this basis the Hamiltonian constraint can be expressed through the dynamical multiplet only,”
Quote in the paperNot a substantial claim - Findingreasoning
The β dependence of the path integral is contained in three multipliers (35)-(37) that cancel each other.
“As a result the dependence of the path integral on β is contained in three multipliers ( 35)-(37) which cancel each other.”
Quote in the paperSubstantial claim
- Findingreasoning
Complex gauge transformations do not destroy the reality of the three-dimensional metric.
“Since {G X , g ij } D = 0, the complex gauge transformations do not destroy the reality of the metric.”
Quote in the paperSubstantial claim - Findingreasoning
The representation is covariant under arbitrary changes of basis in the adjoint representation.
“Moreover, it is covariant under arbitrary transformations of the basis of the adjoint representation.”
Quote in the paperNot a substantial claim - Findingreasoning
A Wilson loop operator can be constructed from the connection as in equation (38).
“Thus the Wilson loop operator can be constructed as”
Quote in the paperSubstantial claim - Findingreasoning
The redefined constraints ̃α are first-class.
“Then ˜ α are first-class constraints with the algebra presented in appendix C.”
Quote in the paperSubstantial claim - Findingnone given
The smeared constraints obey the algebra (C2).
“They obey the following algebra:”
Quote in the paperSubstantial claim - Findingreasoning
The 3+1 decomposed action can be written in an so(4,C)-covariant form.
“As a result we have represented the 3 + 1 decomposed action in the so(4, C) covariant form.”
Quote in the paperSubstantial claim - Findingnone given
In the Dirac bracket {A_Xk, g_ij}_D, the additional contribution cancels the dependence on β.
“i.e. the additional contribution of the Dirac bracket cancels the dependence on β.”
Quote in the paperNot a substantial claim - Findingreasoning
The second-class constraints (12) and (14) are independent of β.
“It is remarkable that the second-class constraints ( 12) and ( 14) do not depend on β that proves consistency of the constraints in different formulations.”
Quote in the paperSubstantial claim - Findingreasoning
The theory depends on only one parameter.
“Besides, the theory depends on one parameter only.”
Quote in the paperNot a substantial claim - Findingreasoning
The fields form multiplets in the adjoint representation of so(4,C).
“As will be shown the fields form multiplets in the adjoint representation of so(4, C) with the following structure constants”
- Also stated in: § 5. Conclusions · plain
“the set of canonical variables forms multiplets in the adjoint representation of this algebra.”
Quote in the paperSubstantial claim - Also stated in: § 5. Conclusions · plain
- Findingnone given
The action S(β) is invariant under the duality transformation A(1) ↔ A(2), β1 ↔ β2.
“The action S (β) is invariant under the 'duality' transformation A (1) ←→ A (2) , β 1 ←→ β 2 .”
Quote in the paperNot a substantial claim - Findingreasoning
A_Xi is a true so(4,C) connection.
“A X i is a true so(4, C)connection.”
- Also stated in: § 4.2. Loop approach · plain
“although A X i is non-commutative due to ( 31) it is transformed as a true connection under the gauge transformations (23).”
Quote in the paperSubstantial claim - Also stated in: § 4.2. Loop approach · plain
- Findingreasoning
The remaining constraints are second class.
“The remaining constraints are second class.”
Quote in the paperSubstantial claim - Findingreasoning
The time evolution of the metric remains real.
“Similarly, the time evolution remains real due to”
Quote in the paperSubstantial claim - Findingreasoning
The operators such as (38) fail to form the loop algebra.
“Due to this the operators such as (38) fail to form the loop algebra.”
Quote in the paperSubstantial claim - Findingreasoning
The Faddeev-Popov determinants in formulations with different β coincide.
“However, it turns out that the determinants in the formulations with different βs coincide.”
Quote in the paperSubstantial claim - Premisecitation
The Immirzi parameter problem arises because the spectra of geometrical operators and black hole entropy are proportional to an unphysical parameter.
“This problem arises due to the results obtained for the spectra of the geometrical operators and the black hole entropy being proportional to an unphysical parameter, which is called 'Immirzi parameter' [7] and appears as a parameter of a canonical transformation in classical gravity [8,9].”
Quote in the paperSubstantial claim0 of 3 sources checked- cites [7] Rovelli C, 1997, Living Rev.
Not checked in the free run.
- cites [8] Rovelli C, 1997
Not checked in the free run.
- cites [9] source
Not checked in the free run.
- Interpretationcitation
The formulation includes Ashtekar-Barbero gravity.
“It encompasses the Ashtekar-Barbero gravity which serves as a basis of quantum loop gravity.”
- Also stated in: § 5. Conclusions · plain
“which encompasses the Ashtekar-Barbero gravity [8].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [8] Rovelli C, 1997
Not checked in the free run.
- Also stated in: § 5. Conclusions · plain
- Premisecitation
At the classical level the Immirzi parameter should be unphysical.
“So at least at the classical level this parameter should be unphysical.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [8] Rovelli C, 1997
Not checked in the free run.
- cites [9] source
Not checked in the free run.
- Premisecitation
The imaginary part of the Ashtekar action vanishes on the surface of the second-class constraints and the Gauss and Lorentz constraints.
“However, in [13] it was established that the imaginary part of the Ashtekar action (2) vanishes on the surface of the second-class constraints and the Gauss and Lorentz constraints.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13] source
Not checked in the free run.
- May combine two claims. 4
- The quote only partly states this claim. 1
- Classified as probably not a substantial claim. 2
May combine two claims.
FindingreasoningIn the covariant formalism there is a unique connection for all values of β, unlike the one-parameter family of connections in Ashtekar-Barbero gravity.
“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.”
Quote in the paperSubstantial claimMay combine two claims.
FindingreasoningIn the Barbero approach the Wick transformation changes the connection rather than the triad, whereas here the connection is unique and there are two triad multiplets.
“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.”
Quote in the paperSubstantial claimThe quote only partly states this claim.
FindingreasoningThe path integral for the theory with action (8) in Yang-Mills gauges is independent of the Immirzi parameter.
“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.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The formal path-integral manipulations are sufficient to establish the β independence of the quantum theory.
May combine two claims.
PremisecitationLoop quantum gravity has yielded discrete spectra for area and volume operators and a derivation of the Bekenstein-Hawking black hole entropy.
“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].”
Quote in the paperSubstantial claim0 of 3 sources checkedProbably not a substantial claim: may not pass S3 (load-bearing).
InterpretationreasoningThe observed duality is trivial because it is only a change of variables.
“Of course, the observed duality is trivial in the sense that it is only a change of variables.”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim. May combine two claims.
PremisecitationHolst showed that in the time gauge the generalized Hilbert-Palatini action reproduces Barbero's formulation and β is the Immirzi parameter.
“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.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [12] Frittelli S, 1996, Class. Quantum Grav. source
Not checked in the free run.
- cites [8] Rovelli C, 1997
Not checked in the free run.
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.
Holst citation points to a different author
MinorThe text attributes the cited result to Holst, but bibliography entry [12] is by Loll. Holst is listed separately as reference [28].
“In [12] Holst has shown that in the 'time' gauge this action reproduces Barbero's formulation [8]”
“12. Loll R 1995 Phys. Rev. Lett. 75 3048”
Immirzi citation points to Lewandowski
MinorThe footnote identifies [9] as a work by Immirzi, whereas bibliography entry [9] names Lewandowski. An Immirzi work is listed as [24].
“† 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].”
“9. Lewandowski J 1995 The operators of quantum gravity Lecture at the Workshop on Canonical and Quantum Gravity (Warsaw)”
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Holst actions for supergravity theories
Kaul, R. K. (2008) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.77.045030 · Cited by 30 · see citing works
First order gravity on the light front
Alexandrov, S. Y., & Simone Speziale, D. (2015) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.91.064043 · Cited by 29 · see citing works
Self-citationTimelike surfaces in Lorentz covariant loop gravity and spin foam models
Alexandrov, S. Y., & Kádár, Z. (2005) · Classical and Quantum Gravity · doi:10.1088/0264-9381/22/17/010 · Cited by 29 · see citing works
Self-citationAchour, J. B., Geiller, M., Noui, K., & Yu, C. (2015) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.91.104016 · Cited by 32 · see citing works
Engle, J., Han, M., & Thiemann, T. (2010) · Cited by 27 · see citing works
A Lorentz-Covariant Connection for Canonical Gravity
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Sachs’ free data in real connection variables
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Reality conditions for Ashtekar gravity from Lorentz-covariant formulation
Alexandrov, S. Y. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/6/002 · Cited by 23 · see citing works
Self-citationAlternative approaches to Lorentz violation invariance in loop quantum gravity inspired models
Alfaro, J., Morales-Técotl, H. A., Reyes, M. C., & Urrutia, L. F. (2004) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.70.084002 · Cited by 21 · see citing works
A note on the Holst action, the time gauge, and the Barbero-Immirzi parameterPreprint
Geiller, M., & Noui, K. (2012) · HAL · Cited by 29 · see citing works
Manifestly Lorentz-covariant variables for the phase space of general relativity
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Towards a Covariant Loop Quantum GravityPreprint
Livine, E. R. (2006) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0608135 · Cited by 17 · see citing works
Spontaneously broken Lorentz symmetry for Hamiltonian gravity
Gielen, S., & Wise, D. K. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.85.104013 · Cited by 16 · see citing works
Immirzi parameter from an external scalar field
Cianfrani, F., & Montani, G. (2009) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.80.084040 · Cited by 16 · see citing works
Lorentz-covariant Hamiltonian analysis of BF gravity with the Immirzi parameter
Celada, M., & Montesinos, M. (2012) · Classical and Quantum Gravity · doi:10.1088/0264-9381/29/20/205010 · Cited by 16 · see citing works
Canonical analysis of Holst action without second-class constraints
Montesinos, M., Romero, J., & Celada, M. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.101.084003 · Cited by 14 · see citing works
Gravity as an SU (1, 1) gauge theory in four dimensions
Liu, H., & Noui, K. (2017) · Classical and Quantum Gravity · doi:10.1088/1361-6382/aa7348 · Cited by 13 · see citing works
Canonical path integral measures for Holst and Plebanski gravity: I. Reduced phase space derivation
Engle, J., Han, M., & Thiemann, T. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/24/245014 · Cited by 13 · see citing works
New "phase" of quantum gravity
Wang, C. H. T. (2006) · CERN Document Server (European Organization for Nuclear Research) · Cited by 12 · see citing works
Hilbert space built over connections with a non-compact structure group
Okołów, A. (2005) · Classical and Quantum Gravity · doi:10.1088/0264-9381/22/7/009 · Cited by 10 · see citing works
Comment on Holst’s Lagrangian formulation
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Gravity asymptotics with topological parameters
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On the counting of black hole states in loop quantum gravityPreprint
Alexandrov, S. Y. (2004) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0408033 · Cited by 12 · see citing works
Self-citationCanonical supergravity with Barbero-Immirzi parameter
Sengupta, S., & Kaul, R. K. (2010) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.81.024024 · Cited by 10 · see citing works
Hamiltonian Gotay-Nester-Hinds analysis of the parametrized unimodular extension of the Holst action
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From Euclidean to Lorentzian loop quantum gravity via a positive complexifier
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New ‘phase’ of quantum gravity
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Spin connection formulations of real Lorentzian general relativity
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Matter in loop quantum gravity without time gauge: A nonminimally coupled scalar field
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Comment on “Immirzi parameter in quantum general relativity”
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The new spin foam models and quantum gravityPreprint
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In Search of Quantum de Sitter Space: Generalizing the Kodama StatePreprint
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SU(1,1) Barbero-like variables derived from Holst action
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Pure Lorentz spin connection theories and uniqueness of general relativity
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Invited review: The new spin foam models and quantum gravity
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Degenerate Plebanski sector and spin foam quantization
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Self-citationRevisiting the solution of the second-class constraints of the Holst action
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Barbero’s formulation from a BF -type action with the Immirzi parameter
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Towards self dual Loop Quantum GravityPreprint
Achour, J. B. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1511.07332 · Cited by 6 · see citing works
A New Perspective on Covariant Canonical GravityPreprint
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Bi-gravity with a single graviton
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Self-citationCanonical structure of minimal varying Λ theories
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Self-citationLinear derivative Cartan formulation of general relativity
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Boost invariance of the gravitational field dynamics: quantization without time gauge
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Canonical Lagrangian Dynamics and General RelativityPreprint
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Immirzi ambiguity, boosts and conformal frames for black holes
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Lorentz-Covariant Loop Quantum Gravity
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Self-citationA discrete discontinuity between the two phases of gravity
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Canonical Lagrangian dynamics and general relativity
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Internal spin angular momentum of an asymptotically flat spacetime
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Translation invariant time-dependent massive gravity: Hamiltonian analysis
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Immirzi ambiguity in the kinematics of quantum general relativity
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A new perspective on covariant canonical gravity
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On the Hamiltonian formalism of the tetrad-connection gravity
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Higher dimensional and supersymmetric extensions of loop quantum gravity
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- 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 › Statistical and Nonlinear 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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