Quantum spin dynamics: VIII. The master constraint
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Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the non -- Lie -- algebra structure of the Dirac algebra of Hamiltonian constraints and was successfully tested in various field theory models. For the case of 3+1 gravity itself, so far only a positive quadratic form for the Master Constraint Operator was derived. In this paper we close this gap and prove that the quadratic form is closable and thus stems from a unique self -- adjoint Master Constraint Operator. The proof rests on a simple feature of the general pattern according to which Hamiltonian constraints in LQG are constructed and thus extends to arbitrary matter coupling and holds for any metric signature. With this result the existence of a physical Hilbert space for LQG is established by standard spectral analysis.
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Recent results indicate that the Barrett-Crane theory appears to fail to yield the correct tensorial structure of the graviton propagator in the low-energy limit.
“Worse, recent results [14] indicate that it appear to fail to yield the correct tensorial structure of the graviton propagator in the low-energy limit [15].”
From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [14]
A richer boundary space is needed for a correct classical limit.
“as well as by the realization that a richer boundary space is needed for a correct classical limit [14].”
From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [14]
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Exploring the diffeomorphism-invariant Hilbert space of a scalar field
Sahlmann, H. (2007) · Classical and Quantum Gravity · doi:10.1088/0264-9381/24/18/003 · Cited by 9 · see citing works
Scalar curvature operator for models of loop quantum gravity on a cubical graph
Lewandowski, J., & Mäkinen, I. (2022) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.106.046013 · Cited by 8 · see citing works
Thiemann, T., & Giesel, K. (2023) · Springer eBooks · doi:10.1007/978-981-19-3079-9_97-1 · Cited by 7 · see citing works
Self-citationThiemann, T. (2024) · General Relativity and Gravitation · doi:10.1007/s10714-024-03313-w · Cited by 7 · see citing works
Self-citation2 + 1 gravity with a positive cosmological constant in LQG: a proposal for the physical state
Pranzetti, D. (2011) · Classical and Quantum Gravity · doi:10.1088/0264-9381/28/22/225025 · Cited by 7 · see citing works
Approximating the physical inner product of Loop Quantum Cosmology
Bahr, B., & Thiemann, T. (2006) · CERN Document Server (European Organization for Nuclear Research) · Cited by 7 · see citing works
Self-citationA discrete, unitary, causal theory of quantum gravity
Wall, A. C. (2013) · Classical and Quantum Gravity · doi:10.1088/0264-9381/30/11/115002 · Cited by 7 · see citing works
Sahlmann, H., & Sherif, W. (2024) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ad7c14 · Cited by 6 · see citing works
Nonperturbative quantum gravity in Fock representations
Thiemann, T. (2024) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.110.124023 · Cited by 6 · see citing works
Self-citationProjective limits of state spaces IV. Fractal label sets
Lanéry, S., & Thiemann, T. (2017) · Journal of Geometry and Physics · doi:10.1016/j.geomphys.2017.08.008 · Cited by 6 · see citing works
Self-citationA path integral for the master constraint of loop quantum gravity
Han, M. (2010) · Classical and Quantum Gravity · doi:10.1088/0264-9381/27/21/215009 · Cited by 6 · see citing works
Loop quantum gravity simplicity constraint as surface defect in complex Chern-Simons theory
Han, M., & Huang, Z. (2017) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.95.104031 · Cited by 6 · see citing works
Approximating the physical inner product of loop quantum cosmology
Bahr, B., & Thiemann, T. (2007) · Classical and Quantum Gravity · doi:10.1088/0264-9381/24/8/011 · Cited by 9 · see citing works
Self-citationGeometrical quantum time in the U(1)3 model of Euclidean quantum gravity
Bakhoda, S., & Ma, Y. (2024) · Communications in Theoretical Physics · doi:10.1088/1572-9494/ad972b · Cited by 4 · see citing works
Master constraint approach to quantum-reduced loop gravity
Mäkinen, I. (2024) · Classical and Quantum Gravity · doi:10.1088/1361-6382/ad4506 · Cited by 3 · see citing works
Deep Learning Spinfoam Vertex Amplitudes: The Euclidean Barrett–Crane Model
Sahlmann, H., & Sherif, W. (2025) · Universe · doi:10.3390/universe11070235 · Cited by 3 · see citing works
Regularized Hamiltonians and Spinfoams
Alesci, E. (2012) · Journal of Physics Conference Series · doi:10.1088/1742-6596/360/1/012041 · Cited by 3 · see citing works
Quantum scalar field in quantum gravity with spherical symmetry
Gambini, R., Pullin, J. A., & Rastgoo, S. (2012) · Journal of Physics Conference Series · doi:10.1088/1742-6596/360/1/012005 · Cited by 1 · see citing works
Thiemann, T., & Giesel, K. (2024) · Springer eBooks · doi:10.1007/978-981-99-7681-2_97 · Cited by 1 · see citing works
Self-citationOn Propagation in Loop Quantum Gravity
Thiemann, T., & Varadarajan, M. (2022) · Universe · doi:10.3390/universe8120615 · Cited by 1 · see citing works
Self-citationCommuting simplicity and closure constraints for 4D spin-foam models
Han, M., & Thiemann, T. (2013) · Classical and Quantum Gravity · doi:10.1088/0264-9381/30/23/235024 · Cited by 1 · see citing works
Self-citationNoui, K. (2009) · Springer eBooks · doi:10.1007/978-90-481-3015-3_19
Gambini, R., & Olmedo, J. (2014) · General Relativity and Gravitation · doi:10.1007/s10714-014-1768-1
Review on hermiticity of the volume operators in Loop Quantum Gravity
Ariwahjoedi, S., Belfaqih, I. H., Sebastian, I., & Zen, F. P. (2019) · General Relativity and Gravitation · doi:10.1007/s10714-019-2541-2
TQFT and Loop Quantum Gravity : 2+1 Theory and Black Hole Entropy
Pranzetti, D. (2011) · theses.fr (ABES) · doi:10.70675/51f92df4zd74bz44c9z95eazf924851241ec
Bonzom, V. (2010) · theses.fr (ABES) · doi:10.70675/248ccc62z7e63z4f22z89ddz3e703205a026
- Authors
- Thiemann, T.
- Corresponding author
- Thiemann, Thomas
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2006-03-14
- Volume
- 23
- Issue
- 7
- Pages
- 2249-2265
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2036644868
- Cited by
- 99
- References
- 48
- Citations by year
- 2025: 1 · 2024: 6 · 2023: 4 · 2022: 2 · 2021: 3 · 2020: 1 · 2019: 1 · 2017: 3 · 2016: 2 · 2015: 3 · 2014: 4 · 2013: 12 · 2012: 7
- FWCI
- 6.82
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Institut Périmètre de physique théorique
- Natural Sciences and Engineering Research Council of Canada
- 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
- loop quantum gravity, Hamiltonian constraint, self-adjoint operators, physical Hilbert space, Dirac algebra, matter coupling, spectral analysis
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