%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: quantum space-times
%%% Primary Category Code: 00.
%%% Filename: QuantumSpaceTimes.tex
%%% Version: 5
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
\begin{document}
This is a fundamental topic on quantum space-times viewed from general relativistic and \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} (QG) standpoints, and includes, for example, \htmladdnormallink{quantum geometry}{http://planetphysics.us/encyclopedia/NAQAT2.html} fundamental notions.
\section{Quantum Space-Times (QST)}
The \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of \emph{quantum space-times (QST)} is fundamental to the development of relativistic \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} and at this point it can only be broadly defined as a \emph{class of mathematical spaces that allow the construction of quantum physical theories in a manner consistent with both relativistic principles and quantum gravity}.
There is no universal agreement amongst either theoretical physicists or mathematicians who \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} on \htmladdnormallink{physical mathematics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} about either a specific definition of such quantum space-times or how to develop a valid \htmladdnormallink{classification}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} theory of quantum space-times. However, several specific definitions or models were proposed and a list of such examples is presented next.
\subsection{Specific Definitions for Models of Quantum Space-Times (QSTs) and Quantum Geometry}
\begin{itemize}
\item QSTs represented by posets or causal sets
\item QSTs represented by so-called quantum topoi with Heyting logic algebra as classifier
\item QSTs represented by \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{quantum field theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} (TQFTs) or \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} \htmladdnormallink{QFTs}{http://planetphysics.us/encyclopedia/HotFusion.html} \item QSTs represented as
\htmladdnormallink{spin foams of spin networks}{http://planetphysics.us/encyclopedia/SpinNetworksAndSpinFoams.html}
\item QSTs represented as a \htmladdnormallink{noncommutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}, algebraic-- and/or ``geometrical''--quantized space as in \emph{\htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}} models for SUSY
\item QSTs represented as generalized Riemannian \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} with quantum \htmladdnormallink{tangent spaces}{http://planetphysics.us/encyclopedia/BilinearMap.html}
\item QSTs represented as presheaves of local nets of \htmladdnormallink{quantum operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} in \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} QFT (AQFT)
\item QSTs represented as \htmladdnormallink{quantum fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} on a (physical) Lattice of geometric points
\item QSTs represented as consisting of quantum loops
\item QSTs represented as fractal dimension spaces
\item QSTs represented as categories or spaces of quantized strings as in \htmladdnormallink{string theories}{http://planetphysics.us/encyclopedia/10DBrane.html}
\item Twistor representations in Quantum Gravity (QG) (introduced by Sir Roger Penrose).
\end{itemize}
\end{document}
