A generalized Hamiltonian constraint operator in loop quantum gravity and its simplest Euclidean matrix elements
Open accessAbstract
We study a generalized version of the Hamiltonian constraint operator in non-perturbative loop quantum gravity. The generalization is based on admitting arbitrary irreducible SU (2) representations in the regularization of the operator, in contrast to the original definition where only the fundamental representation is taken. This leads to a quantization ambiguity and to a family of operators with the same classical limit. We calculate the action of the Euclidean part of the generalized Hamiltonian constraint on trivalent states, using the graphical notation of Temperley-Lieb recoupling theory. We discuss the relation between this generalization of the Hamiltonian constraint and crossing symmetry.
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?
Regularization choices for the scalar constraint have to be made and the result is not unique.
“Regularization choices have to be made and the result is not unique.”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [201]
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
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 · see citing works
Self-citationQuantum 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 · see citing works
Relativistic spin networks and quantum gravity
Barrett, J. W., & Crane, L. (1998) · Journal of Mathematical Physics · doi:10.1063/1.532254 · Cited by 404 · see citing works
Baez, J. C. (1998) · Classical and Quantum Gravity · doi:10.1088/0264-9381/15/7/004 · Cited by 381 · see citing works
ON THE CONSISTENCY OF THE CONSTRAINT ALGEBRA IN SPIN NETWORK QUANTUM GRAVITY
GAMBINI, R., LEWANDOWSKI, J., MAROLF, D., & PULLIN, J. (1998) · International Journal of Modern Physics D · doi:10.1142/s0218271898000103 · Cited by 77 · see citing works
The classical limit and the form of the hamiltonian constraint in nonperturbative quantum gravity
Smolin, L. (1996) · arXiv · doi:10.48550/arxiv.gr-qc/9609034 · Cited by 33 · see citing works
Loop Quantum Gravity and the Meaning of Diffeomorphism InvariancePreprint
Gaul, M., & Rovelli, C. (1999) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9910079 · Cited by 32 · see citing works
Self-citationQuantum Theory of Geometry II: Volume operatorsPreprint
Ashtekar, A., & Lewandowski, J. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9711031 · Cited by 11 · see citing works
Flagged first, then by citations
Research works printed but not linked: 28
- Rovelli C 1998 Loop quantum gravity Living. Rev. Relat. http://www.livingreviews.org/Articles/Volume1/1998- 1rovelli/ (Rovelli C 1997 Preprint gr-qc/9710008) · arXiv
- Brügmann B, Gambini R and Pullin J 1992 Jones polynomials for intersecting knots as physical states of quantum gravity Nucl. Phys. B 385 587 (Brügmann B, Gambini R and Pullin J 1992 Preprint hep-th/9202018) · arXiv
- Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9909063) ( · arXiv
- Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911009) ( · arXiv
- Di Bartolo C, Gambini R, Griego R and Pullin J 1999 Preprint gr-qc/9911010) · arXiv
- Gambini R and Pullin J 2000 Preprint gr-qc/0008031) · arXiv
- Thiemann T 1998 Quantum spin dynamics (QSD) Class. Quantum Grav. 15 839 (Thiemann T 1996 Preprint gr-qc/9606089) · arXiv
- Thiemann T 1998 Quantum spin dynamics (QSD) II Class. Quantum Grav. 15 875 (Thiemann T 1996 Preprint gr-qc/9606090) · arXiv
- Rovelli C and Smolin L 1993 Preprint gr-qc/9308002) · arXiv
- Rovelli C 1995 Preprint gr-qc/9503067) · arXiv
- Gambini R, Lewandowski J, Marolf D and Pullin J 1997 Preprint gr-qc/9710018) · arXiv
- Lewandowski J and Marolf D 1998 Loop constraints: a habitat and their algebra Int. J. Mod. Phys. D 7 299 (Lewandowski J and Marolf D 1997 Preprint gr-qc/9710016) · arXiv
- Thiemann T 2000 Gauge field theory coherent states (GCS): I. General properties Preprint hep-th/0005233 · arXiv
- Thiemann T and Winkler O 2000 Gauge field theory coherent states (GCS) : II. Peakedness properties Preprint hep-th/0005237 · arXiv
- Thiemann T and Winkler O 2000 Gauge field theory coherent states (GCS) : III. Ehrenfest Theorems Preprint hep-th/0005234 · arXiv
- Reisenberger M and Rovelli C 1997 Sum over surfaces form of loop quantum gravity Phys. Rev. D 56 3490 (Reisenberger M and Rovelli C 1996 Preprint gr-qc/9612035) · arXiv
- Baez J C 2000 An introduction to spin foam models of quantum gravity and BF theory Geometry and Quantum Physics (Lecture Notes in Physics 543) ed H Gausterer and H Grosse (Berlin: Springer) p 25 (Baez J C 1999 Preprint gr-qc/9905087) · arXiv
- Borissov R, de Pietri R and Rovelli C 1997 Preprint gr-qc/9703090) · arXiv
- Gaul M and Rovelli C 2000 Loop Quantum Gravity and the meaning of diffeomorphism invariance Towards Quantum Gravity: Proc. 35th Karpacz Int. Winter School on Theoretical Physics (Lecture Notes in Physics 541) ed J Kowalski-Glikman (Berlin: Springer) p 277 (
- de Pietri R and Rovelli C 1996 Preprint gr-qc/9602023) · arXiv
- de Pietri R 1997 On the relation between the connection and the loop representation of quantum gravity Class. Quantum Grav. 14 53 (de Pietri R 1996 Preprint gr-qc/9605064) · arXiv
- Barbero F 1995 Real Ashtekar variables for Lorentzian signature space-times Phys. Rev. D 51 5507 (Barbero F 1994 Preprint gr-qc/9410014) · arXiv
- Lewandowski J 1997 Volume and quantizations Class. Quantum Grav. 14 71 (Lewandowski J 1996 Preprint gr-qc/9602035) · arXiv
- Rovelli C and Smolin L 1994 Preprint gr-qc/9411005) · arXiv
- Kauffman L H and Lins S L 1994 Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (Annals of Mathematics Studies 134) (Princeton, NJ: Princeton University Press)
- Ashtekar A and Lewandowski J 1996 Preprint gr-qc/9602046) · arXiv
- Gaul M 2001 Doctoral Thesis
- Brink D M and Satchler G R 1968 Angular Momentum 2nd edn (Oxford: Clarendon)
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
The Spin-Foam Approach to Quantum Gravity
Perez, A. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-3 · Cited by 677 · see citing works
Bojowald, M. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-11 · Cited by 723 · see citing works
Quantum-reduced loop gravity: Cosmology
Alesci, E., & Cianfrani, F. (2013) · Physical Review D · doi:10.1103/physrevd.87.083521 · Cited by 90 · see citing works
Spin foam quantization and anomalies
Bojowald, M., & Perez, A. (2010) · General Relativity and Gravitation · doi:10.1007/s10714-009-0892-9 · Cited by 35 · see citing works
Finite spin-foam-based theory of three- and four-dimensional quantum gravity
Gambini, R., & Pullin, J. (2002) · Physical Review D · doi:10.1103/physrevd.66.024020 · Cited by 10 · see citing works
Guedes, T. L. M., Mena Marugán, G. A., Müller, M., & Vidotto, F. (2025) · Physical Review D · doi:10.1103/hjdk-kdhk · Cited by 5 · see citing works
- Authors
- Gaul, M., & Rovelli, C.
- Affiliations
- Centre National de la Recherche Scientifique (FR)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2001-04-06
- Volume
- 18
- Issue
- 9
- Pages
- 1593-1624
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2014226715
- Cited by
- 56
- References
- 8
- Citations by year
- 2025: 2 · 2024: 2 · 2023: 1 · 2022: 1 · 2021: 1 · 2020: 1 · 2019: 2 · 2018: 1 · 2017: 1 · 2016: 1 · 2015: 2 · 2014: 2 · 2013: 4 · 2012: 3
- FWCI
- 2.10
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9900791)
- 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
- Hamiltonian constraint operator, loop quantum gravity, SU(2) irreducible representations, quantization ambiguity, crossing symmetry, non-perturbative quantum gravity
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
