Quantum theory of geometry II: Volume operators
Open accessAbstract
A functional calculus on the space of (generalized) connections was recently introduced without any reference to a background metric.It is used to continue the exploration of the quantum Riemannian geometry.Operators corresponding to volume of three-dimensional regions are introduced rigorously.It is shown that there are two natural regularization schemes, each of which leads to a well-defined operator.Both operators can be completely specified by giving their action on states labelled by graphs.The two final results are closely related but differ from one another in that one of the operators is sensitive to the differential structure of graphs at their vertices while the second is sensitive only to the topological characteristics.(The second operator was first introduced by Rovelli and Smolin and De Pietri and Rovelli using a somewhat different framework.)The difference between the two operators can be attributed directly to the standard quantization ambiguity.Underlying assumptions and subtleties of regularization procedures are discussed in detail in both cases because volume operators play an important role in the current discussions of quantum dynamics.
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Symmetric states can be defined as generalized states of quantum geometry and used for reduction to minisuperspace models.
“But symmetric states can be defined as generalized states of quantum geometry [8] which can be used for a reduction to minisuperspace models [9].”
From Absence of a Singularity in Loop Quantum Cosmology (2001) · cites this paper as [8]
The spectrum of the volume operator is discrete.
“The spectrum of the volume operator is also discrete [344, 329, 258, 34].”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [34]
In loop quantum gravity, the kinematic framework has led to physical predictions associated with a polymer-like structure of quantum spacetime.
“This kinematic framework is widely accepted throughout various formulations of the theory, and has led to several physical predictions associated with the 'polymer-like' structure of quantum space-time [4].”
From Nonstandard optics from quantum space-time (1999) · cites this paper as [4]
The Barrett-Crane theory is the best studied model in the 4d euclidean context, based on the vertex amplitude introduced by Barrett and Crane.
“The best studied model in the 4d euclidean context is the Barrett-Crane (BC) theory [8], which is based on the vertex amplitude introduced by Barrett and Crane [7].”
From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [7]
Most derivations of spinfoam models involve peculiar procedures or intuitive and ad hoc steps.
“Indeed, although spinfoam models have been derived in a number of different manners [6, 7, 8, 21], most derivations involve peculiar procedures or intuitive and ad hoc steps.”
From Flipped spinfoam vertex and loop gravity (2008) · cites this paper as [7]
The resulting quantum algebra is exactly the one used in the earlier papers [1-14].
“Furthermore, the resulting quantum algebra is precisely the one used in [1][2][3][4][5][6][7][8][9][10][11][12][13][14].”
From Quantum theory of geometry: III. Non-commutativity of Riemannian structures (1998) · cites this paper as [2]
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- Authors
- Ashtekar, A., & Lewandowski, J.
- Affiliations
- Max Planck Society (DE)
- Venue
- Advances in Theoretical and Mathematical Physics
- Publisher
- International Press of Boston
- Published
- 1997
- Volume
- 1
- Issue
- 2
- Pages
- 388-429
- ISSN
- 1095-0753 (ISSN-L)
- 1095-0761
- OpenAlex
- W2122380701
- Cited by
- 580
- References
- 0
- Citations by year
- 2026: 12 · 2025: 43 · 2024: 21 · 2023: 18 · 2022: 12 · 2021: 16 · 2020: 18 · 2019: 44 · 2018: 18 · 2017: 13 · 2016: 16 · 2015: 18 · 2014: 17 · 2013: 21 · 2012: 25
- FWCI
- 18.66
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Advances in Theoretical and Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Max Planck Institute for Plasma Physics (submittedVersion)
- Max Planck Digital Library (submittedVersion, other-oa)
- MPG.PuRe (Max Planck Society) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9514240)
- Alexander von Humboldt-Stiftung
- University of Pennsylvania
- Pennsylvania State 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 › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Keywords
- volume operator, background-independent quantum gravity, regularization schemes, graph states, topological invariants, quantization ambiguity, quantum dynamics, three-dimensional domains
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