QuantumGravityPrograms 2008-09-28 18:00:08 2008-09-28 18:23:58 Topic quantum gravity research programs % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % 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 \subsection{Quantum Gravity programs} There are several distinct programs aimed at developing a Quantum Gravity theory. These include--but are not limited to-- the following. \begin{itemize} \item $\bullet$ The Penrose, twistors programme applied to an open curved space-time (ref. \cite{Hawking and Penrose}), (which is presumably a globally hyperbolic, relativistic space-time). This may also include the idea of developing a \emph{`sheaf cohomology'} for twistors (ref. \cite {Hawking and Penrose}) 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 $\bullet$ The Weinberg, \emph{supergravity} theory, 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 $\bullet$ The programs of Hawking and Penrose \cite{Hawking and Penrose}) 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 $\bullet$ 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 $\bullet$ 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 $\bullet$ The Baez `categorification' programme (\cite{Baez1}, \cite{Baez2}) 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 $\bullet$ The `monoidal category' and valuation approach initiated by Isham (ref. \cite{Isham1}) to the quantum measurement problem and its possible solution through local-to-global, finite constructions in small categories. \end{itemize} \begin{thebibliography}{99} \bibitem{Baez1} J. Baez. 2004. Quantum quandaries : a category theory perspective, in \emph{Structural Foundations of Quantum Gravity}, (ed. S. French et al.) Oxford Univ. Press. \bibitem{Baez2} J. Baez. 2002. Categorified Gauge Theory. in Proceedings of the Pacific Northwest Geometry Seminar Cascade Topology Seminar,Spring Meeting--May 11 and 12, 2002. University of Washington, Seattle, WA. \bibitem{BGGB05} I.C. Baianu, James Glazebrook, G. Georgescu and Ronald Brown. 2008.``Generalized `Topos' Represntations of Quantum Space-Time: Linking Quantum $N$-Valued Logics with Categories and Higher Dimensional Algebra.'', (\emph{Preprint}) \bibitem{BIsham2} J. Butterfield and C. J. Isham : A topos perspective on the Kochen--Specker theorem I - IV, \emph{Int. J. Theor. Phys}, \textbf{37} (1998) No 11., 2669--2733 \textbf{38} (1999) No 3., 827--859, \textbf{39} (2000) No 6., 1413--1436, \textbf{41} (2002) No 4., 613--639. \bibitem{BIsham1} J. Butterfield and C. J. Isham : Some possible roles for topos theory in quantum theory and quantum gravity, \emph{Foundations of Physics}. \bibitem{FernCastro} F.M. Fernandez and E. A. Castro. 1996. (Lie)\emph{``Algebraic Methods in Quantum Chemistry and Physics.''}, Boca Raton: CRC Press, Inc. \bibitem{Feynman} Feynman, R. P., 1948, ``Space--Time Approach to Non--Relativistic Quantum Mechanics'', \emph{Reviews of Modern Physics}, 20: 367–387. [It is reprinted in (Schwinger 1958).] \bibitem{Hawking and Penrose} S. W. Hawking and R. Penrose. 2000. \emph{The Nature of Space and Time}. Princeton and Oxford: Princeton University Press. \bibitem{PLR} R. J. Plymen and P. L. Robinson: Spinors in Hilbert Space. \emph{Cambridge Tracts in Math.} \textbf{114}, \emph{Cambridge Univ. Press} 1994. \bibitem{Raptis2k} I. Raptis : Algebraic quantisation of causal sets, \emph{Int. Jour. Theor. Phys.} \textbf{39} (2000), 1233. \bibitem{Raptis2} I. Raptis : Quantum space-time as a quantum causal set, arXiv:gr-qc/0201004. \bibitem{noncomm} J. E. Roberts : More lectures on algebraic quantum field theory (in A. Connes, et al. (\emph{Non--commutative Geometry}), Springer (2004). \bibitem{Rovelli} C. Rovelli : Loop quantum gravity (1997), arXiv:gr--qc/9710008. \bibitem{Smit} Jan Smit. 2002. \emph{Quantum Field Theory on a Lattice}. \bibitem{Weinberg} S. Weinberg.1995--2000. \emph{The Quantum Theory of Fields}. Cambridge, New York and Madrid: Cambridge University Press, Vols. 1 to 3. \bibitem{Wess-Bagger} Wess and Bagger. 2000. Supergravity. (Weinberg) \end{thebibliography}