%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: theoretical programs in quantum gravity
%%% Primary Category Code: 04.
%%% Filename: TheoreticalProgramsInQuantumGravity.tex
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\begin{document}
There are several distinct research \htmladdnormallink{programs}{http://planetphysics.us/encyclopedia/SupercomputerArchitercture.html} aimed at developing the mathematical foundations of \htmladdnormallink{quantum gravity theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html}. These include, but are not limited to, the following.
\subsection{Mathematical Programs being Developed in Quantum Gravity}
\begin{enumerate}
\item The twistors program applied to an open curved \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html} (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 \htmladdnormallink{fermion}{http://planetphysics.us/encyclopedia/Fermion.html} of spin-3/2 of undefined \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} and unitary `homogeneity' (which
has not been observed so far);
\item The \emph{\htmladdnormallink{supergravity}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}} theory program, which is consistent with \htmladdnormallink{supersymmetry}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html} and \htmladdnormallink{superalgebra}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html}, and utilizes \emph{graded \htmladdnormallink{Lie algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html}} and \emph{matter-coupled
\htmladdnormallink{superfields}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}} in the presence of \emph{weak} gravitational \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html};
\item The no \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} (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
\htmladdnormallink{metric}{http://planetphysics.us/encyclopedia/MetricTensor.html} with convex \htmladdnormallink{Lorentzian}{http://planetphysics.us/encyclopedia/LebesgueMeasure.html} metrics in the expanding, and then contracting, space-times with
a very small value of \htmladdnormallink{Einstein's}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} \htmladdnormallink{cosmological `constant}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}'. Such Hawking, double-pear shaped,
space-times also have an initial Weyl \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} value close to zero and, ultimately, a largely
fluctuating Weyl tensor during the `final crunch' of our \htmladdnormallink{Universe}{http://planetphysics.us/encyclopedia/MultiVerses.html}, 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 \htmladdnormallink{TQFT/}{http://planetphysics.us/encyclopedia/SUSY2.html} approach that aims at finding the \htmladdnormallink{topological invariants}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} of a
\htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} embedded in an abstract \htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/VectorSpace2.html} related to the \htmladdnormallink{statistical mechanics}{http://planetphysics.us/encyclopedia/ThermodynamicLaws.html} problem of
defining extensions of the partition \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} for many-particle quantum \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html};
\item The string and \htmladdnormallink{superstring}{http://planetphysics.us/encyclopedia/10DBrane.html} theories/M-theory that `live' in higher dimensional
spaces (e.g., $n\geq 6$, preferred $n-dim =11$), and can be considered to be \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of physical entities that vibrate, are quantized, interact, and that might also be able to predict fundamental masses relevant to \htmladdnormallink{quantum particles}{http://planetphysics.us/encyclopedia/QuantumParticle.html};
\item The `\htmladdnormallink{categorification}{http://planetphysics.us/encyclopedia/Categorification3.html}' and \htmladdnormallink{groupoidification}{http://planetphysics.us/encyclopedia/Cod.html} programs (\cite{BAJ-DJ98b,BAJ-DJ2k1}) that aims to deal with \htmladdnormallink{quantum field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} and \htmladdnormallink{QG}{http://planetphysics.us/encyclopedia/SUSY2.html} problems at the abstract level of \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} and \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/Functor.html} in what seems to be mostly a global approach;
\item The `monoidal category' and valuation approach initiated by Isham to the \htmladdnormallink{quantum measurement}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} problem and its possible solution through local-to-global, finite constructions in \htmladdnormallink{small categories}{http://planetphysics.us/encyclopedia/SmallCategory.html}.
\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}
\end{document}
