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Lectures on Non-Perturbative Canonical Gravity
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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“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 the citing paper
“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]
In the citing paper
“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]
In the citing paper
“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]
In the citing paper
“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]
In the citing paper
“In standard quantum cosmology the constraint equation is ˆ c 2 √ | ˆ p | ξ ( c ) = 0 with a solution √ | ˆ p | ξ ( c ) = δ ( c ) which is not unique.”
From Absence of a Singularity in Loop Quantum Cosmology (2001) · cites this paper as [17]
In the citing paper
“This is exactly what one obtains from the classical constraint H (E) = -6 c 2 √ | p | in standard quantum cosmology [17] by quantizing 3ˆ c = -iγl 2 P d / d p . In our framework, however, this is only an approximate equation valid for large scale factors.”
From Absence of a Singularity in Loop Quantum Cosmology (2001) · cites this paper as [17]
In the citing paper
“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 [1]
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Metadata
- 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 › Nuclear and High Energy 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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