Spin networks for noncompact groups
Open accessAbstract
Spin networks are a natural generalization of Wilson loop functionals. They have been extensively studied in the case where the gauge group is compact and it has been shown that they naturally form a basis of gauge invariant observables. Physically the restriction to compact gauge groups is enough for the study of Yang–Mills theories, however it is well known that noncompact groups naturally arise as internal gauge groups for Lorentzian gravity models. In this context, a proper construction of gauge invariant observables is needed. The purpose of the present work is to define the notion of spin network states for noncompact groups. We first build, by a careful gauge fixing procedure, a natural measure and a Hilbert space structure on the space of gauge invariant graph connections. Spin networks are then defined as generalized eigenvectors of a complete set of hermitic commuting operators. We show how the delicate issue of taking the quotient of a space by noncompact groups can be address in term of algebraic geometry. We finally construct the full Hilbert space containing all spin network states. Having in mind applications to gravity, we illustrate our results for the groups SL(2,R) and SL(2,C).
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Recent work on spin networks for noncompact groups allows holonomies that lie in infinite-dimensional unitary representations of the Lorentz group, with a finite inner product defined by appropriate gauge-fixing.
“This new technology essentially allows one to consider holonomies (56) that lie in infinite-dimensional unitary representations of the Lorentz group, with a finite inner product defined by appropriate gauge-fixing.”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [129]
The quantities T0 and T1 can be represented as Hermitian operators on the space of holonomies or on a larger space of spin networks.
“The quantities T0 and T1 can be represented as Hermitian operators on this space of holonomies (or on a larger space of spin networks).”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [129]
Section 5.2 (p. 23) says that the Laplacian operators Δe are Hermitian on a Hilbert space whose eigenvectors are spin networks. Searches for T0, T1, and Hermitian holonomy operators found no passage identifying T0 and T1 or saying they have the claimed representations; the Laplacian result does not establish this claim. (LLM, conf=0.76)
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Cites
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- Authors
- Freidel, L., & Livine, E. R.
- Affiliations
- École Normale Supérieure de Lyon (FR)
- Centre de Physique Théorique (FR)
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 2003-03-01
- Volume
- 44
- Issue
- 3
- Pages
- 1322-1356
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W2044999237
- Cited by
- 60
- References
- 2
- Citations by year
- 2024: 2 · 2023: 1 · 2022: 1 · 2021: 1 · 2020: 1 · 2019: 2 · 2018: 3 · 2017: 3 · 2016: 5 · 2015: 6 · 2014: 4 · 2013: 2 · 2012: 4
- FWCI
- 1.78
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- spin networks, gauge-invariant observables, gauge fixing, Hilbert space construction, algebraic geometry, SL(2,R), SL(2,C), quantum gravity
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