Ashtekar formulation of general relativity and loop-space nonperturbative quantum gravity: a report
Abstract
The formulation of general relativity discovered by Ashtekar (1986, 1987) and the recent results obtained in non-perturbative quantum gravity using loop-space techniques are reviewed. The new formulation is based on the choice of a set of Lagrangian (and Hamiltonian) variables, instead of the spacetime metric. In terms of these new variables, the dynamical equations are remarkably simplified and a structural identity between general relativity and the Yang-Mills theories is revealed. The formalism has proven to be useful in numerous problems in gravitational physics. In quantum gravity, the new formalism has overcome long-standing difficulties and led to unexpected results. A nonperturbative approach to quantum theory has been constructed in terms of the Wilson loops of the Ashtekar connection. This approach, denoted as loop-space representation, has led to the complete solution of the quantum diffeomorphism constraint in terms of knot states, to the discovery of an infinite-dimensional class of solutions to the quantum gravitational dynamics, and to certain surprising indications on the existence of a discrete structure of spacetime around the Planck length. These results are presented in a compact self-contained form. The basic Ashtekar formalism is presented and its applications are outlined. The loop-space representation and the non-perturbative knot states of quantum gravity are described in detail, with particular regard to their physical interpretation and to the information they may provide on the microstructure of spacetime.
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The Poisson bracket in (3) preserves the space of cylindrical functions, and the Dirac delta is absorbed completely by the integrations in the definitions of holonomy and flux.
“The Poisson bracket in (3) preserves the space of cylindrical functions and the Dirac delta is absorbed completely by the integrations involved in the definitions of the holonomy and flux. This fact was pointed out for the first time in [1, 8].”
From Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras (2006) · cites this paper as [8]
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Self-citation
- Authors
- Rovelli, C.
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1991-09-01
- Volume
- 8
- Issue
- 9
- Pages
- 1613-1676
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2134089624
- Cited by
- 221
- References
- 5
- Citations by year
- 2026: 2 · 2025: 27 · 2024: 1 · 2023: 1 · 2022: 3 · 2021: 5 · 2020: 4 · 2019: 4 · 2018: 2 · 2017: 6 · 2016: 2 · 2015: 6 · 2014: 2 · 2013: 3 · 2012: 1
- FWCI
- 20.71
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Classical and Quantum Gravity (publishedVersion)
- 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
- Ashtekar formulation, general relativity, loop quantum gravity, non-perturbative quantum gravity, Wilson loops, discrete spacetime, Planck length
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