non-Abelian Quantum Algebraic TopologyNONABELIANQUANTUMALGEBRAICTOPOLOGY22010-09-28 01:45:052010-10-24 21:01:50Topicquantum algebraic topologyquantum spacetimesquantum state spacenon-AbelianTQFTAQFTnon-Abelian theoryquantum algebraquantum geometryquantum fundamental groupoidquantum algebraalgebraic topologyquantum gravity\section{Non-Abelian Quantum Algebraic Topology (NAQAT)}
This is a new contributed topic (under construction).
\emph{Quantum Algebraic Topology} is the area of theoretical physics and physical mathematics concerned with the applications of Algebraic Topology methods, results and constructions (including its extensions to Category Theory, Topos Theory and Higher Dimensional Algebra) to fundamental quantum physics problems, such as the representations of Quantum spacetimes and Quantum State Spaces in Quantum Gravity, in arbitrary reference frames.
Non--Abelian gauge field theories can also be formalized or presented in the QAT framework.
Perhaps the neighbor areas with which QAT overlaps significantly
are: Algebraic Quantum Field Theories (AQFT)/Local Quantum Physics (LQP), Axiomatic QFT, Lattice QFT (LQFT) and Supersymmetry/Supergravity. One can also claim overlap with various Topological Field Theories (TFT), or Topological Quantum Field Theories (TQFT), Homotopy QFT (HQFT), Dilaton, and Lattice Quantum Gravity (respectively, DQG and LQG) theories.
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\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.}
\med
\med There are several possible applications of the generalized van Kampen theorem in the development of physical representations of a
quantized space-time '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)} which
was precisely described in ref.[69] in terms of \emph{the nerve of an open covering} $N \cU$ of the
topological space $X$ that would be isomorphic to a $k$-simplex $K$ underlying $X$. %%[44]
The corresponding,\emph{noncommutative} algebra $\Omega$ associated
with the finitary $T_0$-poset $P(S)$ is \emph{the Rota algebra}
$\Omega$ discussed in [69], 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 isomorphism) $P(S)$ in the \emph{colimit}, $\lim_\leftarrow
P_N{X}$, recovers a space homeomorphic to $X$ [69]. Other non-Abelian results derived from the generalized van Kampen theorem
were discussed by Brown, Hardie, Kamps and Porter in [72], and by Brown, Higgins and Sivera in [68].
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\subsection{Local--to--Global (LG) Construction Principles
consistent with Quantum `Axiomatics'}
\med A novel approach to QST construction in AQFT may involve the
use of fundamental theorems of algebraic topology generalised from
topological spaces to spaces with structure, such as a filtration,
or as an $n$-cube of spaces. In this category are the generalized,
\emph{Higher Homotopy Seifert-van Kampen theorems (HHSvKT)} of
Algebraic Topology with novel and unique non-Abelian applications.
Such theorems have allowed some new calculations of homotopy types
of topological spaces. They have also allowed new proofs and
generalisations of the classical Relative Hurewicz Theorem by R.
Brown and coworkers [68],[72]. One may find links of such results
to the expected \emph {`non-commutative} geometrical' structure of
quantized space--time [85].
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See also the Exposition on NAQAT at: http://aux.planetphysics.org/files/lec/61/ANAQAT20e.pdf