Background independent quantum gravity: a status report
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The goal of this review is to present an introduction to loop quantum gravity—a background-independent, non-perturbative approach to the problem of unification of general relativity and quantum physics, based on a quantum theory of geometry. Our presentation is pedagogical. Thus, in addition to providing a bird's eye view of the present status of the subject, the review should also serve as a vehicle to enter the field and explore it in detail. To aid non-experts, very little is assumed beyond elements of general relativity, gauge theories and quantum field theory. While the review is essentially self-contained, the emphasis is on communicating the underlying ideas and the significance of results rather than on presenting systematic derivations and detailed proofs. (These can be found in the listed references.) The subject can be approached in different ways. We have chosen one which is deeply rooted in well-established physics and also has sufficient mathematical precision to ensure that there are no hidden infinities. In order to keep the review to a reasonable size, and to avoid overwhelming non-experts, we have had to leave out several interesting topics, results and viewpoints; this is meant to be an introduction to the subject rather than an exhaustive review of it.
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LQG is a proposal for implementing the quantization program established by Dirac, Wheeler, and DeWitt.
“LQG is a proposal for the implementation of the quantization program established in the 1960s by Dirac, Wheeler, and DeWitt, among others (for recent reviews see [35, 356, 323]).”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [35]
In loop quantum gravity, spatial geometrical operators such as area, volume, and length have been constructed on the kinematical Hilbert space and possess discrete spectra.
“One of the celebrated results of loop quantum gravity ͑LQG͒ is the discreteness of the spectrum of geometrical operators such as length, area, and volume operators.”
From Are the spectra of geometrical operators in loop quantum gravity really discrete? (2009) · cites this paper as [2]
The original model of [2,3] and the variant pointed out in [11] appear as limiting γ→0 and γ→∞ cases, respectively.
“The original model of [2, 3] and the variant pointed out in [11] appear as limiting γ → 0 and γ → ∞ cases, respectively.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [11]
All these models are defined by the same vertex, the square of the SU(2) Wigner 15j symbol; they differ for the class of boundary states considered and their measure in the spinfoam sum.
“All these models ([2, 3, 11, 12],) are defined by the same vertex, namely the square of the SU (2) Wigner 15 j symbol; they differ for the class of boundary states considered and their measure in the spinfoam sum.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [11]
The constraints can equivalently be solved using coherent states, yielding the same result as the master-constraint technique.
“In [11], it was shown that these constraints can equivalently be solved using a different technique, based on coherent states, yielding the same result.”
From LQG vertex with finite Immirzi parameter (2008) · cites this paper as [11]
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 [8]
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 [8]
The state ω0 is well known and extensively used in loop quantization.
“The state ω 0 is well known, and is extensively used in the loop quantization [22, 6].”
From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [6]
The semianalytic symmetry group provides a new version of a map Cyl → Cyl* (the algebraic dual) which averages with respect to the allowed diffeomorphisms of Σ.
“For example, the symmetry group provides a new, elegant version of a map Cyl → Cyl ∗ (the algebraic dual) which averages with respect to the (allowed) diffeomorphisms of Σ.”
From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [6]
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Quantum theory of geometry III: Non-commutativity of Riemannian structuresPreprint
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Complexifier Coherent States for Quantum General Relativity
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Refined Algebraic Quantization: Systems with a single constraintPreprint
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Minisuperspaces: Observables and Quantization
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Flagged first, then by citations
Research works printed but not linked: 5
- [26]26. Wisniewski J 2001 Symplectic structure from Holst action Preprint
- [67]67. Lewandowski J, Okolów A, Sahlmann H and Thiemann T 2003 Uniqueness of the diffeomorphism invariant state on the quantum holonomy-flux algebra Preprint Spin networks
- [136]136. R150 Topical Review
- [177]177. Sahlmann H and Thiemann T 2002 Towards the QFT on curved spacetime limit of QGR: II. A concrete implementation Preprint gr-qc/0207031 · arXiv
- [215]215. Okolw A 2004 Representations of quantum geometry PhD Thesis University of Warsaw
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“Finally, as emphasized in section I, a key limitation of the present work is that the theory is not obtained through a systematic symmetry reduction of full LQG.”
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Li, Y., Han, M., Grassl, M., & Zeng, B. (2017) · New Journal of Physics · doi:10.1088/1367-2630/aa7235 · Cited by 15 · see citing works
Two-dimensional topological field theories coupled to four-dimensional BF theory
Montesinos, M., & Perez, A. (2008) · Physical Review D · doi:10.1103/physrevd.77.104020 · Cited by 13 · see citing works
The Quantum Memory Matrix: A Unified Framework for the Black Hole Information Paradox
Neukart, F., Brasher, R., & Marx, E. (2024) · Entropy · doi:10.3390/e26121039 · Cited by 12 · see citing works
Extending the QMM Framework to the Strong and Weak Interactions
Neukart, F., Marx, E., & Vinokur, V. (2025) · Entropy · doi:10.3390/e27020153 · Cited by 11 · see citing works
Guedes, T. L. M., Mena Marugán, G. A., Müller, M., & Vidotto, F. (2025) · Physical Review D · doi:10.1103/hjdk-kdhk · Cited by 5 · see citing works
Multiparticle singlet states cannot be maximally entangled for the bipartitions
Bernards, F., & Gühne, O. (2024) · Journal of Mathematical Physics · doi:10.1063/5.0159105 · Cited by 5 · see citing works
- Authors
- Ashtekar, A., & Lewandowski, J.
- Affiliations
- Pennsylvania State University (US)
- Max Planck Society (DE)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2004-07-10
- Volume
- 21
- Issue
- 15
- Pages
- R53-R152
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2070707213
- Cited by
- 1,951
- References
- 268
- Citations by year
- 2026: 81 · 2025: 145 · 2024: 97 · 2023: 70 · 2022: 64 · 2021: 75 · 2020: 68 · 2019: 79 · 2018: 62 · 2017: 72 · 2016: 97 · 2015: 68 · 2014: 91 · 2013: 88 · 2012: 107
- FWCI
- 64.56
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Max Planck Institute for Plasma Physics (submittedVersion)
- Max Planck Digital Library (submittedVersion)
- MPG.PuRe (Max Planck Society) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (0090091)
- Alexander von Humboldt-Stiftung
- 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 › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- loop quantum gravity, background independence, quantum geometry, unification of quantum mechanics and general relativity, non-perturbative quantum gravity
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