Talk:PlanetPhysics/Non Abelian Quantum Algebraic Topology 2

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: non-Abelian Quantum Algebraic Topology %%% Primary Category Code: 00. %%% Filename: NONABELIANQUANTUMALGEBRAICTOPOLOGY2.tex %%% Version: 20 %%% 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}

\section{Non-Abelian Quantum Algebraic Topology (NAQAT)}

This is a new contributed topic (under construction).

\emph{Quantum Algebraic Topology} is the area of \htmladdnormallink{theoretical physics}{http://planetphysics.us/encyclopedia/NonNewtonian2.html} and \htmladdnormallink{physical mathematics}{http://planetphysics.us/encyclopedia/NonNewtonian2.html} concerned with the applications of \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} methods, results and constructions (including its extensions to \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{topos}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html} Theory and \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html}) to fundamental quantum physics problems, such as the \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of Quantum spacetimes and Quantum State Spaces in \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html}, in arbitrary \htmladdnormallink{reference frames}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html}. Non--Abelian gauge \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} theories can also be formalized or presented in the \htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/QATs.html} framework.

Perhaps the neighbor areas with which QAT overlaps significantly are: \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{quantum field theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} (AQFT\htmladdnormallink{)/local quantum physics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} (LQP), Axiomatic QFT, Lattice \htmladdnormallink{QFT}{http://planetphysics.us/encyclopedia/HotFusion.html} (\htmladdnormallink{LQFT}{http://planetphysics.us/encyclopedia/LQG2.html}) and \htmladdnormallink{supersymmetry/}{http://planetphysics.us/encyclopedia/Supersymmetry.html}. One can also claim overlap with various \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} Field Theories (TFT), or Topological Quantum Field Theories (TQFT), \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} QFT (\htmladdnormallink{HQFT}{http://planetphysics.us/encyclopedia/NonNewtonian2.html}), Dilaton, and Lattice Quantum Gravity (respectively, DQG and \htmladdnormallink{LQG}{http://planetphysics.us/encyclopedia/LQG2.html}) theories.

\subsection{Applications of the Van Kampen Theorem to Crossed Complexes. Representations of Quantum Space-Time in terms of Quantum Crossed Complexes over a Quantum Groupoid.}

There are several possible applications of the \htmladdnormallink{generalized Van Kampen theorem}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} in the development of physical representations of a quantized \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html} `geometry' For example, a possible application of the generalized van Kampen theorem is the construction of the initial, quantized space-time as the \emph{unique colimit} of \emph{quantum causal sets (posets)} in terms of \emph{the nerve of an open \htmladdnormallink{covering}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}} $N \textbf{U}$ of the topological space $X$ that would be isomorphic to a $k$-simplex $K$ underlying $X$. The corresponding,\emph{\htmladdnormallink{noncommutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}} algebra $\Omega$ associated with the finitary $T_0$-poset $P(S)$ is \emph{the Rota algebra} $\Omega$, and the \emph{quantum topology} $T_0$ is defined by the partial ordering arrows for regions that can overlap, or superpose, coherently (in the quantum sense) with each other. When the poset $P(S)$ contains $2N$ points we write this as $P_{2N}(S)$. The \emph{unique} (up to an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}) $P(S)$ in the \emph{colimit}, $\lim_\leftarrow P_N{X}$, recovers a space homeomorphic to $X$. Other non-Abelian results derived from the generalized van Kampen theorem were discussed by Brown, Hardie, Kamps and Porter, and also by Brown, Higgins and Sivera.

\subsection{Local--to--Global (LG) Construction Principles consistent with Quantum Axiomatics}

A novel approach to \htmladdnormallink{QST}{http://planetphysics.us/encyclopedia/SUSY2.html} construction in AQFT may involve the use of fundamental \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} of algebraic topology generalised from topological spaces to spaces with structure, such as a filtration, or as an $n$-cube of spaces. In this \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} are the generalized, \emph{\htmladdnormallink{higher homotopy}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} Seifert-van Kampen theorems (HHSvKT)} of Algebraic Topology with novel and unique non-Abelian applications. Such theorems have allowed some new calculations of homotopy \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of topological spaces. They have also allowed new proofs and generalisations of the classical \htmladdnormallink{relative Hurewicz theorem}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} by R. Brown and coworkers. One may find links of such results to the expected \emph \htmladdnormallink{http://planetphysics.us/encyclopedia/AbelianCategory3.html} geometrical structure of quantized space--time.

See also the Exposition on NAQAT at: http://aux.planetphysics.org/files/lec/61/ANAQAT20e\htmladdnormallink{.pdf}{http://planetphysics.us/encyclopedia/LebesgueMeasure.html}

\end{document}