Separable Hilbert space in loop quantum gravity
Open accessAbstract
We study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphism-invariant states is nonseparable. This is a consequence of the fact that the knot space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the fine-tuning of the mathematical setting. We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable.
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The Rovelli-Fairbairn generalized diffeomorphisms are defined to be smooth everywhere except a finite set of points, and therefore they would not be useful in our case.
“The Rovelli-Fairbairn generalized diffeomorphisms, however, are defined to be smooth everywhere except a finite set of points, therefore they would not be useful in our case.”
From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [11]
There are several similar, non-equivalent extensions of the analytic diffeomorphisms recently introduced in the literature.
“There are several similar, non-equivalent extension of the analytic diffeomorphisms recently introduced in literature.”
From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [11]
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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 · cited 1× · see citing works
“In gravity, the continuous dependence of position is gauged away by diffeomorphism invariance, dramatically reducing the size of the resulting state space [10, 12].”
BackgroundSelf-citationIntroduction to Modern Canonical Quantum General RelativityPreprint
Thiemann, T. (2001) · arXiv (Cornell University) · doi:10.48550/arxiv.gr-qc/0110034 · Cited by 449 · see citing works
A combinatorial approach to diffeomorphism invariant quantum gauge theories
Zapata, J. A. (1997) · Journal of Mathematical Physics · doi:10.1063/1.532159 · Cited by 32 · see citing works
Combinatorial Space from Loop Quantum Gravity
Zapata, J. A. (1998) · General Relativity and Gravitation · doi:10.1023/a:1026699012787 · Cited by 31 · see citing works
Flagged first, then by citations
Research works printed but not linked: 4
- [1][1] V.I. Arnold, Catastrophe Theory (Springer-Verlag, Berlin, 1984). -
- [2][2] J.C. Baez, J. Muniain, Gauge Fields, Knots and Gravity (World Scientific, 1994). -
- [5][5] N.D. Gilbert and T. Porter, Knots and Surfaces (Oxford Univiversity Press, Oxford 1994). -
- [9][9] C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2004) to appear; a draft of the book can be found in http://www.cpt.univ-mrs.fr/ ∼ rovelli . -
Cited by
Sorted by citations
Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras
Lewandowski, J., Okołów, A., Sahlmann, H., & Thiemann, T. (2006) · Communications in Mathematical Physics · doi:10.1007/s00220-006-0100-7 · Cited by 297 · see citing works
Quantum spin dynamics: VIII. The master constraint
Thiemann, T. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/7/003 · Cited by 99 · see citing works
Particle Scattering in Loop Quantum Gravity
Modesto, L., & Rovelli, C. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.191301 · Cited by 93 · see citing works
Self-citation
- Authors
- Fairbairn, W., & Rovelli, C.
- Affiliations
- Centre de Physique Théorique (FR)
- Centre de Physique Théorique de Luminy, Case 907, F-13288 Marseille, France
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 2004-06-09
- Volume
- 45
- Issue
- 7
- Pages
- 2802-2814
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W2075550086
- Cited by
- 68
- References
- 4
- Citations by year
- 2026: 2 · 2024: 2 · 2023: 5 · 2022: 1 · 2021: 2 · 2018: 1 · 2017: 2 · 2016: 4 · 2015: 1 · 2014: 6 · 2013: 3 · 2012: 3
- FWCI
- 5.66
- 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)
- 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
- loop quantum gravity, separability, kinematical Hilbert space, diffeomorphism invariance, knot spaces, moduli, countability
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