TheoreticalProgramsInQuantumGravity 2009-02-02 04:21:39 2009-02-02 04:21:39 Topic theoretical and mathematical approaches to quantum gravity % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \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{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} There are several distinct research programs aimed at developing the mathematical foundations of quantum gravity theories. These include, but are not limited to, the following. \subsection{Mathematical programs developed in quantum gravity} \begin{enumerate} \item The twistors program applied to an open curved space-time (see refs. \cite{SH2k4, RP2k}), (which is presumably a globally hyperbolic, relativistic space-time). This may also include the idea of developing a \emph{`sheaf cohomology'} for twistors (see ref. \cite{RP2k}) but still needs to justify the assumption in this approach of a charged, fundamental fermion of spin-3/2 of undefined mass and unitary `homogeneity' (which has not been observed so far); \item The \emph{supergravity} theory program, which is consistent with supersymmetry and superalgebra, and utilizes \emph{graded Lie algebras} and \emph{matter-coupled superfields} in the presence of \emph{weak} gravitational fields; \item The no boundary (closed), \emph{continuous} space-time programme (ref. \cite{SH2k4}) in quantum cosmology, concerned with singularities, such as black and `white' holes; S. W. Hawking combines, joins, or glues an initially flat Euclidean metric with convex Lorentzian metrics in the expanding, and then contracting, space-times with a very small value of Einstein's cosmological `constant'. Such Hawking, double-pear shaped, space-times also have an initial Weyl tensor value close to zero and, ultimately, a largely fluctuating Weyl tensor during the `final crunch' of our universe, presumed to determine the irreversible arrow of time; furthermore, an observer will always be able to access through measurements only \emph{a limited part} of the global space-times in our universe; \item The TQFT/HQFT approach that aims at finding the topological invariants of a manifold embedded in an abstract vector space related to the statistical mechanics problem of defining extensions of the partition function for many-particle quantum systems; \item The string and superstring theories/M-theory that `live' in higher dimensional spaces (e.g., $n\geq 6$, preferred $n-dim =11$), and can be considered to be topological representations of physical entities that vibrate, are quantized, interact, and that might also be able to predict fundamental masses relevant to quantum particles; \item The `categorification' and groupoidification programs (\cite{BAJ-DJ98b,BAJ-DJ2k1}) that aims to deal with quantum field and QG problems at the abstract level of categories and functors in what seems to be mostly a global approach; \item The `monoidal category' and valuation approach initiated by Isham to the quantum measurement problem and its possible solution through local-to-global, finite constructions in small categories. \end{enumerate} \begin{thebibliography}{9} \bibitem{SH2k4} S.Hawkings. 2004. \emph{The beginning of time}. \bibitem{RP2k} R. Penrose. 2000. {Shadows of the mind.}, Cambridge University Press: Cambridge, UK. \bibitem{BAJ-DJ98b} Baez, J. and Dolan, J., 1998b, \emph{``Categorification'', Higher Category Theory, Contemporary Mathematics}, \textbf{230}, Providence: \emph{AMS}, 1-36. \bibitem{BAJ-DJ2k1} Baez, J. and Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in \emph{Mathematics Unlimited -- 2001 and Beyond}, Berlin: Springer, pp. 29--50. \end{thebibliography}