Loop representations for 2+1 gravity on a torus
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We study the loop representation of the quantum theory for 2+1-dimensional general relativity on a manifold M=T 2 *R, where T 2 is the torus, and compare it with the connection representation for this system. In particular, we look at the loop transform in the part of the phase space where the holonomies are boosts, and study its kernel. This kernel is dense in the connection representation, and the transform is not continuous with respect to the natural topologies, even in its domain of definition. Nonetheless, loop representations isomorphic to the connection representation corresponding to this part of the phase space can still be constructed if due care is taken. We present this construction, but note that certain ambiguities remain; in particular, functions of loops cannot be uniquely associated with functions of connections.
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In the spacelike sector, the loop transform is not well-behaved: a dense set of Chern-Simons states transforms to zero.
“In this sector, the transform is not very well-behaved: In fact, a dense set of Chern-Simons states transforms to zero [192].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [192]
For the torus universe, the loop transform can be obtained explicitly and written as a simple integral transform.
“For the torus universe, this “loop transform” can be obtained explicitly [26, 29, 192], and written as a simple integral transform.”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [192]
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Representations of the holonomy algebras of gravity and nonAbelian gauge theories
Ashtekar, A., & Isham, C. J. (1992) · Classical and Quantum Gravity · doi:10.1088/0264-9381/9/6/004 · Cited by 310 · see citing works
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Research works printed but not linked: 5
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Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
- Authors
- Marolf, D. M.
- Affiliations
- Syracuse University (US)
- Corresponding author
- Marolf, D M
- ORCID
- Marolf, D M
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1993-12-01
- Volume
- 10
- Issue
- 12
- Pages
- 2625-2647
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W1971026774
- Cited by
- 37
- References
- 1
- Citations by year
- 2024: 1 · 2019: 2 · 2016: 1 · 2015: 1 · 2013: 1 · 2012: 1
- FWCI
- 5.38
- 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)
- Funders
- National Science Foundation (9005790)
- Syracuse 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 representation, torus, holonomy, quantum gravity, phase space
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