LOOP CONSTRAINTS: A HABITAT AND THEIR ALGEBRA
Open accessAbstract
This work introduces a new space [Formula: see text] of 'vertex-smooth' states for use in the loop approach to quantum gravity. Such states provide a natural domain for Euclidean Hamiltonian constraint operators of the type introduced by Thiemann (and using certain ideas of Rovelli and Smolin). In particular, such operators map [Formula: see text] into itself, and so are actual operators in this space. Their commutator can be computed on [Formula: see text] and compared with the classical hypersurface deformation algebra. Although the classical Poisson bracket of Hamiltonian constraints yields an inverse metric times an infinitesimal diffeomorphism generator, and despite the fact that the diffeomorphism generator has a well-defined nontrivial action on [Formula: see text], the commutator of quantum constraints vanishes identically for a large class of proposals.
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The loss of spacetime interpretation for the SU(2) connection is probably the reason why the quantum constraint algebra does not reproduce the classical one and contains an anomaly.
“In particular, this fact is probably the reason why the quantum constraint algebra does not reproduce the classical one and contains an anomaly [18, 19, 17].”
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- Authors
- LEWANDOWSKI, J., & MAROLF, D.
- Affiliations
- Max Planck Institute for Gravitational Physics (DE)
- Syracuse University (US)
- Venue
- International Journal of Modern Physics D
- Publisher
- World Scientific Pub Co Pte Lt
- Published
- 1998-04
- Volume
- 07
- Issue
- 02
- Pages
- 299-330
- ISSN
- 0218-2718 (ISSN-L)
- 1793-6594
- OpenAlex
- W2025714345
- Cited by
- 76
- References
- 11
- Citations by year
- 2025: 1 · 2021: 1 · 2019: 1 · 2018: 3 · 2017: 2 · 2016: 4 · 2015: 3 · 2014: 5 · 2013: 7 · 2012: 8
- FWCI
- 3.99
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- International Journal of Modern Physics D (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Max Planck Institute for Plasma Physics (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
- loop quantum gravity, Hamiltonian constraint operator, Poisson brackets, constraint algebra
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