The physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory
Open accessAbstract
A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In this paper we reconsider the implementation of the constraints that defines the model. We define in a simple way the boundary Hilbert space of the theory, introducing a slight modification of the embedding of the SU(2) representations into the SL(2,C) ones. We then show directly that all constraints vanish on this space in a weak sense. The vanishing is exact (and not just in the large quantum number limit.) We also generalize the definition of the volume operator in the spinfoam model to the Lorentzian signature, and show that it matches the one of loop quantum gravity, as does in the Euclidean case.
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?
Geometric operators such as volume and area defined in the covariant context coincide with the corresponding operators in the canonical formulation.
“Moreover, geometric operators such as volume and area defined in the covariant context are shown to coincide with the corresponding operators defined in the canonical formulation [141, 142].”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [141]
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
Kiefer, C. (2004) · Cited by 2,981 · see citing works
Rovelli, C. (2004) · doi:10.1017/cbo9780511755804 · Cited by 2,240 · see citing works
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
General relativity without coordinates
Regge, T. (1961) · Il Nuovo Cimento · doi:10.1007/bf02733251 · Cited by 1,477 · see citing works
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 · cited 1× · see citing works
“The kinematics of canonical loop quantum gravity is rather well understood; in particular, the properties of the geometrical operators, including the area and the volume operators[27, 28, 29] are well established.”
BackgroundQuantum 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
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 · see citing works
Representations of the Rotation and Lorentz Groups and Their Applications.
Hermann, R., Gelfand, I. M., Minlos, R. A., & Shapiro, Z. Y. (1966) · American Mathematical Monthly · doi:10.2307/2315370 · Cited by 784 · see citing works
Spin networks and quantum gravity
Rovelli, C., & Smolin, L. (1995) · Physical Review D · doi:10.1103/physrevd.52.5743 · Cited by 629 · see citing works
LQG vertex with finite Immirzi parameter
Engle, J., Livine, E., Pereira, R., & Rovelli, C. (2008) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2008.02.018 · Cited by 584 · see citing works
Quantum theory of geometry II: Volume operators
Ashtekar, A., & Lewandowski, J. (1997) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.1997.v1.n2.a8 · Cited by 580 · see citing works
Spin foam models for quantum gravity
Perez, A. (2003) · Classical and Quantum Gravity · doi:10.1088/0264-9381/20/6/202 · Cited by 532 · see citing works
A new spin foam model for 4D gravity
Freidel, L., & Krasnov, K. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/12/125018 · Cited by 521 · see citing works
An Introduction to Spin Foam Models of BF Theory and Quantum Gravity
Baez, J. C. (2000) · Lecture Notes in Physics · doi:10.1007/3-540-46552-9_2 · Cited by 464 · see citing works
Knot Theory and Quantum Gravity
Rovelli, C., & Smolin, L. (1988) · Physical Review Letters · doi:10.1103/physrevlett.61.1155 · Cited by 463 · see citing works
Introduction to Modern Canonical Quantum General RelativityPreprint
Thiemann, T. (2001) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0110034 · Cited by 449 · see citing works
Relativistic Spin Networks and Quantum GravityPreprint
Barrett, J. W., Crane, L., & Crane, J. W. B. L. (1998) · Cited by 423 · 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
New spinfoam vertex for quantum gravity
Livine, E. R., & Speziale, S. (2007) · Physical Review D · doi:10.1103/physrevd.76.084028 · Cited by 401 · see citing works
Weaving a classical metric with quantum threads
Ashtekar, A., Rovelli, C., & Smolin, L. (1992) · Physical Review Letters · doi:10.1103/physrevlett.69.237 · Cited by 399 · 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
“Sum over surfaces” form of loop quantum gravity
Reisenberger, M. P., & Rovelli, C. (1997) · Physical Review D · doi:10.1103/physrevd.56.3490 · Cited by 372 · see citing works
Loop-Quantum-Gravity Vertex Amplitude
Engle, J., Pereira, R., & Rovelli, C. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.161301 · Cited by 335 · see citing works
A Lorentzian Signature Model for Quantum General RelativityPreprint
Barrett, J. W., & Crane, L. (2000) · Cited by 282 · see citing works
A Lorentzian signature model for quantum general relativity
Barrett, J. W., & Crane, L. (2000) · Classical and Quantum Gravity · doi:10.1088/0264-9381/17/16/302 · Cited by 275 · see citing works
FUNDAMENTAL STRUCTURE OF LOOP QUANTUM GRAVITY
Han, M., Ma, Y., & HUANG, W. (2007) · International Journal of Modern Physics D · doi:10.1142/s0218271807010894 · Cited by 274 · see citing works
Barrett–Crane model from a Boulatov–Ooguri field theory over a homogeneous space
De Pietri, R., Freidel, L., Krasnov, K., & Rovelli, C. (2000) · Nuclear Physics B · doi:10.1016/s0550-3213(00)00005-5 · Cited by 244 · see citing works
Flipped spinfoam vertex and loop gravity
Engle, J., Pereira, R., & Rovelli, C. (2008) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2008.02.002 · Cited by 241 · see citing works
Spacetime as a Feynman diagram: the connection formulation
Reisenberger, M. P., & Rovelli, C. (2001) · Classical and Quantum Gravity · doi:10.1088/0264-9381/18/1/308 · Cited by 202 · see citing works
Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity
Oriti, D. (2001) · Reports on Progress in Physics · doi:10.1088/0034-4885/64/12/203 · Cited by 201 · see citing works
The Lorentz group and harmonic analysis
Rühl, W. (1970) · CERN Document Server (European Organization for Nuclear Research) · Cited by 188 · see citing works
Three-dimensional loop quantum gravity: physical scalar product and spin-foam models
Noui, K., & Perez, A. (2005) · Classical and Quantum Gravity · doi:10.1088/0264-9381/22/9/017 · Cited by 181 · see citing works
Solving the simplicity constraints for spinfoam quantum gravity
Livine, E. R., & Simone Speziale, D. (2008) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/81/50004 · Cited by 180 · see citing works
A new look at loop quantum gravity
Carlo Rovelli, C. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/11/114005 · Cited by 162 · see citing works
Eugenio Bianchi, A., Carlo Rovelli, C., & Vidotto, F. (2010) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.82.084035 · Cited by 158 · see citing works
AN INVITATION TO LOOP QUANTUM GRAVITY
Smolin, L. (2004) · International Symposium on Quantum Theory and Symmetries (QTS) · doi:10.1142/9789812702340_0078 · Cited by 155 · see citing works
A spin foam model without bubble divergences
Perez, A., & Rovelli, C. (2001) · Nuclear Physics B · doi:10.1016/s0550-3213(01)00030-x · Cited by 113 · see citing works
An Introduction to Loop Quantum Gravity Through CosmologyPreprint
Ashtekar, A. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0702030 · Cited by 107 · see citing works
Casting loop quantum cosmology in the spin foam paradigm
Ashtekar, A., Campiglia, M., & Henderson, A. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/13/135020 · Cited by 102 · see citing works
Generalized Functions. Volume 5: Integral Geometry and Representation Theory
Gelfand, I. M., Граев, М. И., & Vilenkin, N. Y. (1967) · Cited by 94 · see citing works
Loop quantum cosmology and spin foams
Ashtekar, A., Campiglia, M., & Henderson, A. (2009) · Physics Letters B · doi:10.1016/j.physletb.2009.10.042 · Cited by 81 · see citing works
Lorentzian loop quantum gravity vertex amplitude
Roberto Pereira, B. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/8/085013 · Cited by 77 · see citing works
Projected spin networks for Lorentz connection: linking spin foams and loop gravity
Livine, E. R. (2002) · Classical and Quantum Gravity · doi:10.1088/0264-9381/19/21/316 · Cited by 77 · see citing works
Projected Spin Networks for Lorentz connection: Linking Spin Foams and Loop GravityPreprint
Livine, E. R. (2002) · Cited by 74 · see citing works
A lattice worldsheet sum for 4-d Euclidean general relativityPreprint
Reisenberger, M. (1997) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9711052 · Cited by 73 · see citing works
The face amplitude of spinfoam quantum gravity
Eugenio Bianchi, A., Regoli, D., & Carlo Rovelli, C. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/18/185009 · Cited by 72 · see citing works
Worldsheet formulations of gauge theories and gravityPreprint
Reisenberger, M. (1994) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/9412035 · Cited by 60 · see citing works
Gluing 4-simplices: A derivation of the Barrett-Crane spin foam model for Euclidean quantum gravity
Oriti, D., & Williams, R. M. (2000) · Physical Review D · doi:10.1103/physrevd.63.024022 · Cited by 54 · see citing works
SPIN FOAMS AS FEYNMAN DIAGRAMS
Reisenberger, M., & Carlo Rovelli, C. (2003) · Johns Hopkins Workshop on Current Problems in Particle Theory (JHW) · doi:10.1142/9789812791368_0023 · Cited by 51 · see citing works
New vertices and canonical quantization
Alexandrov, S. (2010) · Physical Review D · doi:10.1103/physrevd.82.024024 · Cited by 48 · see citing works
LQG vertex with finite Immirzi parameter
Engle, J., Livine, E. R., Pereira, R., et al. (2007) · HAL (Le Centre pour la Communication Scientifique Directe) · doi:10.48550/arxiv.0711.0146 · Cited by 46 · see citing works
The EPRL intertwiners and corrected partition function
Kamiński, W., Kisielowski, M., & Lewandowski, J. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/16/165020 · Cited by 43 · see citing works
Consistently Solving the Simplicity Constraints for Spinfoam Quantum GravityPreprint
Livine, E. R., & Speziale, S. (2007) · arXiv (Cornell University) · doi:10.48550/arxiv.0708.1915 · Cited by 37 · see citing works
On the spinfoam expansion in cosmology
Carlo Rovelli, C., & Vidotto, F. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/14/145005 · Cited by 36 · see citing works
Triangulated loop quantum cosmology: Bianchi IX universe and inhomogeneous perturbations
Battisti, M. V., Marcianò, A., & Carlo Rovelli, C. (2010) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.81.064019 · Cited by 35 · see citing works
Coherent states, constraint classes and area operators in the new spin-foam models
Engle, J., & Roberto Pereira, B. (2008) · Classical and Quantum Gravity · doi:10.1088/0264-9381/25/10/105010 · Cited by 32 · see citing works
The volume operator in covariant Quantum GravityPreprint
Ding, Y., & Carlo Rovelli, C. (2010) · Cited by 31 · see citing works
Self-citationSemiclassical analysis of the Loop Quantum Gravity volume operator: I. Flux Coherent StatesPreprint
Flori, C., & Thiemann, T. (2008) · arXiv (Cornell University) · doi:10.48550/arxiv.0812.1537 · Cited by 30 · see citing works
Geometries Induced by Surfaces: an Algorithm on the 3-Dimensional SIMPLEX Lattice
Iwasaki, J. (1996) · magr · Cited by 2 · see citing works
Flagged first, then by citations
Research works printed but not linked: 1
- [17][17] R Pereira, 'Lorentzian LQG vertex amplitude' , Class. Quant. Grav. 25 (2008) 085013; arXiv:0710.5043 [gr-qc] - · arXiv
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 · see citing works
Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory
Rovelli, C., & Vidotto, F. (2014) · doi:10.1017/cbo9781107706910 · Cited by 544 · see citing works
A new look at loop quantum gravity
Carlo Rovelli, C. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/11/114005 · Cited by 162 · see citing works
Edge modes of gravity. Part III. Corner simplicity constraints
Freidel, L., Geiller, M., & Pranzetti, D. (2021) · Journal of High Energy Physics · doi:10.1007/jhep01(2021)100 · Cited by 109 · see citing works
4-dimensional spin-foam model with quantum Lorentz group
Han, M. (2011) · Journal of Mathematical Physics · doi:10.1063/1.3606592 · Cited by 98 · see citing works
Carlo Rovelli, C. (2013) · doi:10.22323/1.140.0003 · Cited by 109 · see citing works
Loop quantum gravity: the first 25 years
Carlo Rovelli, C. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/15/153002 · Cited by 124 · see citing works
The Continuum Limit of Loop Quantum Gravity: A Framework for Solving the Theory
Dittrich, B. (2017) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789813220003_0006 · Cited by 91 · see citing works
Lorentz covariance of loop quantum gravity
Rovelli, C., & Speziale, S. (2011) · Physical Review D · doi:10.1103/physrevd.83.104029 · Cited by 84 · see citing works
Spin Foams and Canonical Quantization
Alexandrov, S. Y. (2012) · Symmetry Integrability and Geometry Methods and Applications · doi:10.3842/sigma.2012.055 · Cited by 93 · see citing works
Lifting SU(2) spin networks to projected spin networks
Dupuis, M., & Livine, E. R. (2010) · Physical Review D · doi:10.1103/physrevd.82.064044 · Cited by 71 · see citing works
Lorentzian spinfoam propagator
Eugenio Bianchi, A., & Ding, Y. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.86.104040 · Cited by 70 · see citing works
Self-citationDing, Y., Han, M., & Rovelli, C. (2011) · Physical Review D · doi:10.1103/physrevd.83.124020 · Cited by 60 · see citing works
Self-citationGroup Field Theory Condensate Cosmology: An Appetizer
Pithis, A. G. A., & Sakellariadou, M. (2019) · Universe · doi:10.3390/universe5060147 · Cited by 54 · see citing works
In quantum gravity, summing is refining
Carlo Rovelli, C., & Smerlak, M. (2012) · Classical and Quantum Gravity · doi:10.1088/0264-9381/29/5/055004 · Cited by 45 · see citing works
From General Relativity to Quantum Gravity
Ashtekar, A., Reuter, M., & Carlo Rovelli, C. (2015) · Cambridge University Press eBooks · doi:10.1017/cbo9781139583961.016 · Cited by 45 · see citing works
Complex Ashtekar Variables and Reality Conditions for Holst’s Action
Wieland, W. (2011) · Annales Henri Poincaré · doi:10.1007/s00023-011-0134-z · Cited by 43 · see citing works
Critical overview of loops and foams
Alexandrov, S. Y., & Roche, P. (2011) · Physics Reports · doi:10.1016/j.physrep.2011.05.002 · Cited by 42 · see citing works
Magliaro, E., & Perini, C. (2013) · International Journal of Modern Physics D · doi:10.1142/s0218271813500016 · Cited by 39 · see citing works
Emergence of gravity from spinfoams
Magliaro, E., & Perini, C. (2011) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/95/30007 · Cited by 38 · see citing works
Thiemann complexifier in classical and quantum FLRW cosmology
Achour, J. B., & Livine, E. R. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.96.066025 · Cited by 32 · see citing works
Simple model for quantum general relativity from loop quantum gravity
Carlo Rovelli, C. (2011) · Journal of Physics Conference Series · doi:10.1088/1742-6596/314/1/012006 · Cited by 38 · see citing works
Complex Ashtekar variables and reality conditions for Holst’s actionPreprint
Wieland, W. (2016) · Cited by 34 · see citing works
Twistorial phase space for complex Ashtekar variablesPreprint
Wieland, W. (2016) · Cited by 23 · see citing works
Vertex expansion for the Bianchi type I model
Campiglia, M., Henderson, A., & Nelson, W. (2010) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.82.064036 · Cited by 22 · see citing works
The Construction of Spin Foam Vertex Amplitudes
Eugenio Bianchi, A. (2013) · Symmetry Integrability and Geometry Methods and Applications · doi:10.3842/sigma.2013.008 · Cited by 16 · see citing works
Wieland, W. (2013) · Classical and Quantum Gravity · doi:10.1088/0264-9381/31/2/025002 · Cited by 15 · see citing works
Graphical calculus of volume, inverse volume and Hamiltonian operators in loop quantum gravity
Yang, J., & Ma, Y. (2017) · The European Physical Journal C · doi:10.1140/epjc/s10052-017-4713-0 · Cited by 15 · see citing works
Magliaro, E., & Perini, C. (2012) · International Journal of Modern Physics D · doi:10.1142/s0218271812500903 · Cited by 13 · see citing works
Commuting Simplicity and Closure Constraints for 4D Spin Foam Models
Han, M., & Thiemann, T. (2013) · Max Planck Digital Library · Cited by 13 · see citing works
The new spin foam models and quantum gravityPreprint
Pérez, A. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1205.0911 · Cited by 9 · see citing works
Spinfoam Models for Quantum Gravity
Livine, E. R. (2024) · Encyclopedia of Mathematical Physics · doi:10.1016/b978-0-323-95703-8.00253-6 · Cited by 8 · see citing works
Invited review: The new spin foam models and quantum gravity
Pérez, A. (2012) · Papers in Physics · doi:10.4279/pip.040004 · Cited by 15 · see citing works
Quantum geometric maps and their properties
Finocchiaro, M., Jeong, Y., & Oriti, D. (2021) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ac0c30 · Cited by 7 · see citing works
Complex Ashtekar variables, the Kodama state and spinfoam gravityPreprint
Wieland, W. (2011) · arXiv (Cornell University) · doi:10.48550/arxiv.1105.2330 · Cited by 7 · see citing works
Asymptotic of Lorentzian Polyhedra PropagatorPreprint
Puchta, J. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1307.4747 · Cited by 4 · see citing works
Semi-Classical Holomorphic Transition Amplitudes in Covariant Loop Quantum GravityPreprint
D’Ambrosio, F. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2001.04651 · Cited by 3 · see citing works
Li, H., Han, M., Liu, H., Song, S., & Qu, D. (2025) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.111.086017 · Cited by 1 · see citing works
New insights in quantum geometry
Sahlmann, H. (2012) · Journal of Physics Conference Series · doi:10.1088/1742-6596/360/1/012007
Ashtekar, A., Berger, B. K., Isenberg, J. A., & MacCallum, M. A. H. (2015) · Cambridge University Press eBooks · doi:10.1017/cbo9781139583961.014
- Authors
- Ding, Y., & Rovelli, C.
- Affiliations
- Centre de Physique Théorique (FR)
- ORCID
- Ding, You
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2010-09-07
- Volume
- 27
- Issue
- 20
- Pages
- 205003
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2014405245
- Cited by
- 42
- References
- 57
- Citations by year
- 2025: 1 · 2024: 1 · 2021: 2 · 2020: 1 · 2019: 2 · 2017: 3 · 2016: 2 · 2015: 2 · 2014: 2 · 2013: 7 · 2012: 7
- FWCI
- 5.90
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion)
- HAL AMU (submittedVersion, other-oa)
- Trinity's Access to Research Output (TARA) (Trinity College Dublin) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Agence Nationale de la Recherche (ANR-09-BLAN-0041)
- 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
- spin foam models, volume operator, quantum gravity, discrete spacetime
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
