QuantizedRiemannSpaces 2008-10-03 15:35:51 2008-10-03 15:35:51 Topic quantized Riemann spaces noncommutative geometry % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \setlength{\textwidth}{6.5in} %\setlength{\textwidth}{16cm} \setlength{\textheight}{9.0in} %\setlength{\textheight}{24cm} \hoffset=-.75in %%ps format %\hoffset=-1.0in %%hp format \voffset=-.4in \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\med}{\medbreak} \newcommand{\medn}{\medbreak \noindent} \newcommand{\bign}{\bigbreak \noindent} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \section{Quantized Riemannian Manifolds and Geometry} An interesting, but perhaps limiting approach to Quantum Gravity (QG), involves defining a \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). Whereas a classical Riemannian manifold has a metric defined by a special, \emph{Riemannian tensor}, the \emph{quantum Riemannian geometry} may be defined in different theoretical approaches to QG by either quantum loops (or perhaps `strings'), or \emph{spin networks and spin foams} (in locally covariant GR quantized space-times). The latter two concepts are related to the `standard' quantum spin observables and thus have the advantage of precise mathematical definitions. As spin foams can be defined as functors of spin network categories, \emph{quantized space-times (QST)s} can be represented by, or defined in terms of, \emph{natural transformations of `spin foam' functors}. The latter definition is not however the usual one adopted for quantum Riemannian geometry, and other (for example, noncommutative geometry) approaches attempt to define a QST metric not by a \emph{Riemannian tensor} --as in the classical GR case-- but in relation to a generalized, quantum `Dirac' operator in a spectral triplet. \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. \begin{definition} \emph{Quantum Geometry} is defined as a \emph{field of Mathematical or Theoretical Physics based on geometrical and Algebraic Topology approaches to Quantum Gravity}- one such approach is based on Noncommutative Geometry and SUSY (the `Standard' Model in current Physics). \end{definition} \emph{A Result for Quantum Spin Foam Representations of Quantum Space-Times (QST)s:} There exists an $n$-connected CW model $(Z,QSF)$ for the pair $(QST,QSF)$ such that: $f_*: \pi_i(Z) \rightarrow \pi_i (QST)$, is an isomorphism for $i>n$, and it is a monomorphism for $i=n$. The $n$-connected CW model is unique up to homotopy equivalence. (The $CW$ complex, $Z$, considered here is a homotopic `hybrid' between QSF and QST). \begin{thebibliography}{9} \bibitem{AC94} A. Connes. 1994. \emph{Noncommutative Geometry}. Academic Press: New York and London. \bibitem{AL2k5} Abhay Ashtekar and Jerzy Lewandowski.2005. Quantum Geometry and Its Applications. \PMlinkexternal{PDF file download}{http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf}. \end{thebibliography}