quantum Riemannian geometry
QuantumRiemannianGeometry
2008-10-19 23:36:26
2008-10-19 23:36:26
Topic
quantum gravity theories, non-commutative geometry, non-Abelian geometry, non-Abelian topology, noncommutative topology
a mathematical approach to quantum gravity based on noncommutative geometry/noncommutative operator algebra
Quantum Gravity Theories
Non-Commutative Geometry
Non-Abelian Geometry
Non-Abelian Topology
Noncommutative Topology
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\textbf{Description:}
\emph{Quantum geometry (or quantum geometries)} is an approach (resp. are approaches) to Quantum Gravity that are based on either noncommutative geometry and SUSY (the `Standard' Model of current Physics) \cite{AC94,CA85} 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.
\textbf{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 \emph{`quantum' Riemannian geometry} \cite{AL2k5} in place of the classical Riemannian manifold that is employed in the well-known, Einstein's classical approach to General Relativity (GR).
\begin{thebibliography}{9}
\bibitem{AC94}
A. Connes. 1994. \emph{Noncommutative Geometry}. Academic Press: New York and London.
\bibitem{CA85}
Connes, A. 1985 .Non-commutative differential geometry I--II.
\emph{Publication Math\'ematiques IHES}, {\bf 62}, 41--144.
\bibitem{AL2k5}
Abhay Ashtekar and Jerzy Lewandowski. 2005. Quantum Geometry and Its Applications.
\PMlinkexternal{Available PDF download}{http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf}.
\end{thebibliography}