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Cited by 1,663Relativity and Gravitational TheoryNoncommutative and Quantum Gravity TheoriesBlack Holes and Theoretical Physics

New Variables for Classical and Quantum Gravity

Ashtekar, Abhay·1986·Physical Review Letters
DOI 10.1103/physrevlett.57.2244
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Abstract

A Hamiltonian formulation of general relativity based on certain spinorial variables is introduced. These variables simplify the constraints of general relativity considerably and enable one to imbed the constraint surface in the phase space of Einstein's theory into that of Yang-Mills theory. The imbedding suggests new ways of attacking a number of problems in both classical and quantum gravity. Some illustrative applications are discussed.

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  • A solution to problem 1 has been suggested: one can define real Ashtekar variables which simplify the rescaled Hamiltonian constraint of Euclidean gravity and then construct a Wick rotation transform from the Euclidean to the Lorentzian regime where complex Ashtekar variables are needed to simplify the Lorentzian constraint.

    “A solution to problem 1) has been suggested in [4] (see also [5]) : namely, one can define real Ashtekar variables [6] which simplify the rescaled Hamiltonian constraint of Euclidean gravity and then construct a Wick rotation transform from the Euclidean to the Lorentzian regime where the complex Ashtekar variables are needed to simplify the Lorentzian constraint.”

    From Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity (1996) · cites this paper as [5]

  • LQG and spin foams are based on the possibility of formulating Hamiltonian general relativity in terms of Yang-Mills-like connection variables.

    “LQG and spin foams are based on the possibility of formulating Hamiltonian general relativity in terms of Yang-Mills-like connection variables.”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [23]

  • A spin foam picture emerges from considering the time evolution of spin networks.

    “A spin foam picture emerges when considering the evolution in time of spin networks [16,17], which were discovered to represent states of quantum general relativity at the kinematical level [18][19][20][21][22], from the canonical side.”

    From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [17]

  • The kinematical framework developed in this paper is applied in [22], which extends the dynamical framework of [16,17,18,19,20] to the standard model coupled to gravity and other kind of matter.

    “The kinematical framework developed in this paper is applied in [22] which extends the dynamical framework of [16, 17, 18, 19, 20] to the standard model coupled to gravity and other kind of matter.”

    From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [16]

  • For the Einstein-Dirac theory in variables (A^i_a, E^a_i), the symplectic structure for each fermionic degree of freedom is Θ = i/2 ∫ d^3x √det(q)[η ̇η - ̇ηη] with scalar-valued η.

    “The second reason becomes evident when recalling that for the Einstein-Dirac theory, formulated in terms of the variables ( A i a , E a i ) [22] which have proved to be successful for quantizing the Wheeler-DeWitt constraint [16, 17], the symplectic structure for each of the fermionic degrees of freedom is given by Θ = i 2 ∫ Σ d 3 x √ det( q )[ η ˙ η -˙ ηη ] with scalar-valued η .”

    From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [16]

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Authors
Ashtekar, A.
Affiliations
University of California, Santa Barbara (US)
Corresponding author
Ashtekar, Abhay
ORCID
Ashtekar, Abhay
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
1986-11-03
Volume
57
Issue
18
Pages
2244-2247
ISSN
0031-9007 (ISSN-L)
DOI
10.1103/physrevlett.57.2244
OpenAlex
W2098989214
Cited by
1,663
References
0
Citations by year
2026: 39 · 2025: 95 · 2024: 58 · 2023: 49 · 2022: 35 · 2021: 69 · 2020: 43 · 2019: 55 · 2018: 37 · 2017: 43 · 2016: 54 · 2015: 53 · 2014: 58 · 2013: 59 · 2012: 37
FWCI
23.02
Type
journal article
Language
en
Open access
closed
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • PubMed (publishedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
general relativity, Hamiltonian formulation, constraints, Yang-Mills theory, constraint surface, phase space, quantum gravity, classical gravity

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