Description: Quantum geometry (or quantum geometries) is an approach (resp. 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).
References
[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.
