%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: CW-complex of spin networks (CWSN)
%%% Primary Category Code: 03.
%%% Filename: CWComplexOfSpinNetworksCWSN.tex
%%% Version: 2
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\begin{definition}
A \emph{$CW$ complex}, denoted as $X_c$, is a special \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space ($X$) which is the \emph{\htmladdnormallink{union}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html}} of an expanding sequence of subspaces $X^n$, such that, inductively, the first member of this expansion sequence is $X^0$ -- a discrete set of points called the \emph{vertices} of $X$, and $X^{n+1}$ is the \emph{\htmladdnormallink{pushout}{http://planetphysics.us/encyclopedia/Pushout.html}} obtained from $X^n$ by attaching disks $D^{n+1}$ along ``attaching maps'' $j: S^n \rightarrow X^n$. Each resulting map $D^{n+1} \longrightarrow X$ is called a \emph{cell}. (The subscript ``$c$'' in $X_c$, stands for the fact that this (CW) type of topological space $X$ is called \emph{cellular}, or ``made of cells''). The subspace $X^n$ is called the ``$n$-skeleton'' of $X$.
Pushouts, expanding sequence and unions are here understood in the topological sense, with the compactly generated
topologies (\emph{viz.} p.71 in P. J. May, 1999 \cite{MJP1999}).
\end{definition}
\textbf{Examples of a $CW$ complex}:
\begin{enumerate}
\item A \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Cod.html} is a one--dimensional $CW$ complex.
\item \emph{\htmladdnormallink{spin networks}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}} are represented as graphs and they are therefore also one--dimensional $CW$ complexes.
The transitions between \emph{spin networks} lead to \emph{\htmladdnormallink{spin foams}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}}, and spin foams may be thus regarded
as a higher dimensional $CW$ complex (of dimension $d \geq 2$).
\end{enumerate}
\emph{Note.}
The \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of {\em spin networks} and {\em spin foams} were recently developed in the context
of \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} as part of the more general effort of attempting to formulate mathematically a concept of \emph{\htmladdnormallink{quantum state space}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html}} which is also applicable, or relates to \emph{\htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html}} \htmladdnormallink{spacetimes}{http://planetphysics.us/encyclopedia/SR.html}. The {\em \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html}}-- which is fundamental in quantum theories-- has no corresponding concept in \htmladdnormallink{classical mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}. (However, classical \emph{momenta} (both linear and angular) have corresponding \htmladdnormallink{quantum observable}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} that are quite different in form, with their eigenvalues taking on different sets of values in \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} than the ones that might be expected from classical mechanics for the `corresponding' classical observables); the spin is an \emph{intrinsic} observable of all massive \htmladdnormallink{quantum `particles',}{http://planetphysics.us/encyclopedia/QuantumParticle.html} such as electrons, protons, \htmladdnormallink{neutrons}{http://planetphysics.us/encyclopedia/Pions.html}, atoms, as well as of all \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} quanta, such as photons, \emph{\htmladdnormallink{gravitons}{http://planetphysics.us/encyclopedia/BoseEinsteinStatistics.html}}, \htmladdnormallink{gluons}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html}, and so on; furthermore, every quantum `particle' has also associated with it a de Broglie \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}, so that it cannot be realized, or `pictured', as any kind of classical `body'. For massive quantum particles such as electrons, protons, neutrons, atoms, and so on, the spin property has been initially observed for atoms by applying a \htmladdnormallink{magnetic field}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html} as in the famous Stern-Gerlach experiment, (although the applied field may also be electric or gravitational, (see for example \cite{WH52})). All such spins interact with each other thus giving rise to ``spin networks'', which can be mathematically represented as in the second example above; in the case of electrons, protons and neutrons such interactions are magnetic dipolar ones, and in an over-simplified, but not a physically accurate `picture', these are often thought of as `very tiny magnets--or magnetic dipoles--that line up, or flip up and down together, etc'.
\begin{remark}
An earlier, alternative definition of CW complex is also in use that may have
advantages in certain applications where the concept of pushout might not be apparent; on the other hand
as pointed out in \cite{MJP1999} the \textbf{Definition 0.1} presented here has advantages in proving
results, including generalized, or extended \htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html} in
\htmladdnormallink{Algebraic Topology}{http://planetmath.org/?op=getobj&from=lec&id=73},
(as for example in \cite{MJP1999}).
\end{remark}
\begin{thebibliography}{99}
\bibitem{MJP1999}
May, J.P. 1999, \emph{A Concise Course in Algebraic Topology.}, The University of Chicago Press: Chicago.
\bibitem{CRFM1980}
C.R.F. Maunder. 1980, \htmladdnormallink{Algebraic Topology.}{http://planetmath.org/?op=getobj&from=books&id=181},
Dover Publications, Inc.: Mineola, New York.
\bibitem{JJR1998}
Joseph J. Rothman. 1998,
\htmladdnormallink{An Introduction to Algebraic Topology}{http://planetmath.org/?op=getobj&from=books&id=172},
Springer-Verlag: Berlin
\bibitem{WH52}
Werner Heisenberg. {\em The Physical Principles of Quantum Theory}. New York: Dover Publications, Inc.(1952), pp.39-47.
\bibitem{BF92}
F. W. Byron, Jr. and R. W. Fuller. {\em Mathematical Principles of Classical and Quantum Physics.}, New York: Dover Publications, Inc. (1992).
\end{thebibliography}
\end{document}
