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Cited by 872Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

Lectures on Non-Perturbative Canonical Gravity

Ashtekar, Abhay·1991·Advanced Series in Astrophysics and Cosmology
DOI 10.1142/1321
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Abstract

Notes prepared in Collaboration with Ranjeet S Tate It is now generally recognized that perturbative field theoretical methods that have been highly successful in the quantum description of non-gravitational interactions cannot be used as a means of constructing a quantum theory of gravity. The primary aim of the book is to present an up- to-date account of a non-perturbative, canonical quantization program for gravity. Many of the technical results obtained in the process are of interest also to differential geometry, classical general relativity and QCD. The program as a whole was highlighted in virtually every major conference in gravitational physics over the past three years.

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  • In 2+1 dimensions, using the loop representation, one can recast the final description in a purely combinatorial fashion, at least in the time-like sector.

    “Now, in 2+1 dimensions, using the loop representation one can recast the final description in a purely combinatorial fashion (at least in the so-called 'time-like sector' of the theory).”

    From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [31]

  • In 2+1 dimensions with Λ=0, loop quantum gravity is closely related to the first order formalism of Section 2.3 but takes loop variables T0[γ] and T1[γ] as fundamental observables.

    “In 2 + 1 dimensions with Λ = 0, this approach is closely related to the first order formalism of Section 2.3, but takes as its fundamental observables the loop variables T0[γ] and T1[γ] of Equation (55).”

    From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [26]

  • Loop quantum gravity starts with a Hamiltonian formulation of the first order formalism, with constraints (63) written in analogy to the (3+1)-dimensional case.

    “More precisely, loop quantum gravity starts with a Hamiltonian formulation of the first order formalism, with constraints, written in analogy to the (3 + 1)-dimensional case [26], that take the form 63”

    From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [26]

  • For the torus universe, the loop transform can be obtained explicitly and written as a simple integral transform.

    “For the torus universe, this “loop transform” can be obtained explicitly [26, 29, 192], and written as a simple integral transform.”

    From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [26]

  • For the torus universe, a complete, explicit description of loop representation states is possible.

    “The loop representation is complicated by the existence of Mandelstam identities [190] among holonomies of loops, but for the case of the torus universe, a complete, explicit description of the states is again possible [26, 29].”

    From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [26]

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Authors
Ashtekar, A.
Affiliations
Syracuse University (US)
Corresponding author
Ashtekar, Abhay
ORCID
Ashtekar, Abhay
Venue
Advanced Series in Astrophysics and Cosmology
Publisher
WORLD SCIENTIFIC
Published
1991-07
ISSN
1793-1312 (ISSN-L)
DOI
10.1142/1321
OpenAlex
W1673177227
Cited by
872
References
0
Citations by year
2026: 7 · 2025: 25 · 2024: 8 · 2023: 9 · 2022: 16 · 2021: 19 · 2020: 16 · 2019: 29 · 2018: 21 · 2017: 26 · 2016: 21 · 2015: 20 · 2014: 18 · 2013: 16 · 2012: 35
FWCI
14.99
Type
book
Language
en
Open access
closed
Locations
  • Advanced series in astrophysics and cosmology (publishedVersion)
  • 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 › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
quantum gravity, canonical gravity, general relativity, quantum field theory of gravity, differential geometry, classical general relativity, QCD

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