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# quantum geometry

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).

# 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.

## Mathematics Subject Classification

18D25*no label found*18-00

*no label found*81-00

*no label found*81P05

*no label found*81Q05

*no label found*

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