Spectra of length and area in (2 1) Lorentzian loop quantum gravity
Open accessAbstract
We study the spectrum of the length and area operators in Lorentzian loop quantum gravity, in 2+1 spacetime dimensions. We find that the spectrum of spacelike intervals is continuous, whereas the spectrum of timelike intervals is discrete. This result contradicts the expectation that spacelike intervals are always discrete. On the other hand, it is consistent with the results of the spin foam quantization of the same theory.
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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 [130]
Section II (p. 5) constructs SO(2,1) spin networks using holonomies in infinite-dimensional unitary representations and describes an orthonormal basis of gauge-invariant states. It does not establish the attributed detail that an appropriate gauge-fixing defines a finite inner product; its displayed scalar-product construction concerns compact groups. (LLM, conf=0.82)
Experimental — accuracy not yet measuredSpin network states can be defined in 2+1 dimensions as well, and there has been recent work on their role as kinematic states.
“Such states can be defined in 2 + 1 dimensions as well, and there has been some interesting recent work on their role as “kinematic” states [130].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [130]
The appearance of both continuous and discrete labels has a physical interpretation: continuous representations describe spacelike edges and imply a continuous length spectrum, while discrete representations label timelike edges and suggest discrete time.
“The appearance of both continuous and discrete labels has a nice physical interpretation [130]: Continuous representations describe spacelike edges, and seem to imply a continuous length spectrum, while discrete representations label timelike edges, and suggest discrete time.”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [130]
Freidel et al. argue that spacelike intervals are continuous, while timelike intervals are discrete, with a spectrum of the form sqrt(n(n-1)) lP.
“Freidel et al. argue that spacelike intervals are continuous, while timelike intervals are discrete, with a spectrum of the form”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [130]
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 [130]
Section II discusses holonomy states, spin networks, and Hermitian operators representing the frame field. But our copy does not identify quantities T0 and T1 or say that those quantities have Hermitian representations on either space. Searches for T0/T1 and the claimed operator formulation found no decisive passage; page numbers were unavailable. (LLM, conf=0.74)
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Cites
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 · see citing works
Self-citationQuantum 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
<i>SO</i> (4, <i>C</i> )-covariant Ashtekar-Barbero gravity and the Immirzi parameter
Alexandrov, S. (2000) · Classical and Quantum Gravity · doi:10.1088/0264-9381/17/20/307 · Cited by 112 · see citing works
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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
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
Are the spectra of geometrical operators in loop quantum gravity really discrete?
Dittrich, B., & Thiemann, T. (2009) · Journal of Mathematical Physics · doi:10.1063/1.3054277 · Cited by 96 · see citing works
Hyperinvariant spin network states — an AdS/CFT model from first principles
Otto, F., Mansuroglu, R., Schuch, N., Gühne, O., & Sahlmann, H. (2026) · Journal of High Energy Physics · doi:10.1007/jhep06(2026)145 · Cited by 1 · see citing works
- Authors
- Freidel, L., Livine, E. R., & Rovelli, C.
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2003-03-24
- Volume
- 20
- Issue
- 8
- Pages
- 1463-1478
- ISSN
- 0264-9381 (ISSN-L)
- OpenAlex
- W2045777935
- Cited by
- 93
- References
- 3
- Citations by year
- 2026: 1 · 2025: 18 · 2024: 4 · 2023: 1 · 2022: 1 · 2021: 2 · 2020: 2 · 2019: 4 · 2018: 4 · 2017: 2 · 2016: 1 · 2015: 3 · 2014: 6 · 2013: 4 · 2012: 3
- FWCI
- 4.36
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (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 › 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, 2+1 dimensions, length spectrum, area spectrum, discrete spacetime, quantum geometry
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