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Description: Quantum geometry (or quantum geometries) are approaches to Quantum Gravity based on either noncommutative geometry and SUSY (the `Standard' Model of current Physics) [1,2] or modified or `deformed' Riemannian, `quantum' geometry, with additional assumptions regarding a generalized `Dirac' operator, the `spectral triplet' with non-Abelian structures of quantized space-times.
Remarks. Other approaches to Quantum Gravity include: Loop Quantum Gravity (LQG), AQFT approaches, Topological Quantum Field Theory (TQFT)/ Homotopy Quantum Field Theories (HQFT; Tureaev and Porter, 2005), Quantum Theories on a Lattice (QTL), string theories and spin network models.
An interesting, but perhaps limiting approach, involves `quantum' Riemannian geometry [3] in place of the classical Riemannian manifold that is employed in the well-known, Einstein's classical approach to General Relativity (GR).
- 1
- A. Connes. 1994. Noncommutative Geometry. Academic Press: New York and London.
- 2
- Connes, A. 1985 .Non-commutative differential geometry I-II. Publication Mathématiques IHES, 62, 41-144.
- 3
- Abhay Ashtekar and Jerzy Lewandowski. 2005. Quantum Geometry and Its Applications. Available PDF download.
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"quantum geometry" is owned by bci1.
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See Also: noncommutative geometry, quantum gravity theories, quantum Riemannian geometry
| Other names: |
Non-Commutative Geometry, Non-Abelian Geometry, Non-Abelian Topology, Noncommutative Topology |
| Also defines: |
a mathematical approach to quantum gravity based on noncommutative geometry/noncommutative operator algebra |
| Keywords: |
Quantum Gravity Theories, Non-Commutative Geometry, Non-Abelian Geometry, Non-Abelian Topology, Noncommutative Topology |
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Cross-references: Riemannian manifold, place, geometry, theories, string, lattice, quantum theories, homotopy, TQFT, quantum field theory, loop, non-abelian structures, triplet, operator, current, Quantum Gravity
There are 6 references to this entry.
This is version 9 of quantum geometry, born on 2008-08-01, modified 2008-10-17.
Object id is 10897, canonical name is QuantumGeometry.
Accessed 710 times total.
Classification:
| AMS MSC: | 81Q05 (Quantum theory :: General mathematical topics and methods in quantum theory :: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations) | | | 81P05 (Quantum theory :: Axiomatics, foundations, philosophy :: General and philosophical) | | | 81-00 (Quantum theory :: General reference works ) | | | 18-00 (Category theory; homological algebra :: General reference works ) | | | 18D25 (Category theory; homological algebra :: Categories with structure :: Strong functors, strong adjunctions) |
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Pending Errata and Addenda
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