Talk:PlanetPhysics/Quantum Riemannian Geometry

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\textbf{Description:} \emph{Quantum geometry (or quantum geometries)} is an approach (resp. are approaches) to \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} based on either \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} and \htmladdnormallink{SUSY}{http://planetphysics.us/encyclopedia/SUSY4.html} (the `Standard' Model of current Physics) \cite{AC94,CA85} or modified or `deformed' Riemannian, `quantum' geometry, with additional assumptions regarding a generalized `Dirac' \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}, the `spectral triplet' with \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} structures of quantized \htmladdnormallink{space-times}{http://planetphysics.us/encyclopedia/SR.html}.

\textbf{Remarks.} Other approaches to Quantum Gravity include: Loop Quantum Gravity (\htmladdnormallink{LQG}{http://planetphysics.us/encyclopedia/GCGR.html}), \htmladdnormallink{AQFT}{http://planetphysics.us/encyclopedia/PureState.html} approaches, \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{quantum field theory}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} (\htmladdnormallink{TQFT}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html})/ \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} Quantum Field Theories (\htmladdnormallink{HQFT}{http://planetphysics.us/encyclopedia/QAT.html}; Tureaev and Porter, 2005), \htmladdnormallink{quantum theories on a lattice}{http://planetphysics.us/encyclopedia/FTNIR.html} (\htmladdnormallink{QTL}{http://planetphysics.us/encyclopedia/FTNIR.html}), \htmladdnormallink{string theories}{http://planetphysics.us/encyclopedia/10DBrane.html} and \htmladdnormallink{spin network}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html} models. \\

An interesting, but perhaps limiting approach, involves \emph{`quantum' Riemannian geometry} \cite{AL2k5} in place of the classical Riemannian \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} that is employed in the well-known, \htmladdnormallink{Einstein's}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} classical approach to General Relativity (\htmladdnormallink{GR}{http://planetphysics.us/encyclopedia/SR.html}).

\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. \htmladdnormallink{Available PDF download}{http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf}. \end{thebibliography}

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