View source for PlanetPhysics/Spin Networks Viewed As CW Complexes

===Physical Background===
The [[../PreciseIdea/|concepts]] of ''spin networks''  and ''spin foams''  were recently developed in the context of [[../PhysicalMathematics2/|mathematical physics]] as part of the more general effort of attempting to formulate mathematically a concept of ''[[../QuantumSpinNetworkFunctor2/|quantum state space]]'' which is also applicable, or relates to ''[[../LQG2/|quantum gravity]]'' [[../SR/|spacetimes]]. The ''\htmladdnormallink{spin'' {http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} [[../QuantumSpinNetworkFunctor2/|observable]]}-- which is fundamental in quantum theories-- has no corresponding concept in [[../MathematicalFoundationsOfQuantumTheories/|classical mechanics]]. (However, classical ''momenta''  (both linear and angular) have corresponding [[../QuantumOperatorAlgebra5/|quantum observable]] [[../QuantumOperatorAlgebra4/|operators]] that are quite different in form, with their eigenvalues taking on different sets of values in [[../QuantumParadox/|quantum mechanics]] than the ones that might be expected from classical mechanics for the `corresponding' classical observables); the spin is an ''intrinsic''  observable of all massive [[../QuantumParticle/|quantum `particles',]] such as electrons, protons, [[../Pions/|neutrons]], atoms, as well as of all [[../CosmologicalConstant/|field]] quanta, such as photons, ''[[../BoseEinsteinStatistics/|gravitons]]'', [[../ExtendedQuantumSymmetries/|gluons]], and so on; furthermore, every quantum `particle' has also associated with it a de Broglie [[../CosmologicalConstant2/|wave]], 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 [[../NeutrinoRestMass/|magnetic field]] as in the famous Stern-Gerlach experiment, (although the applied field may also be electric or gravitational, (see for example <ref name='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'.

===Spin Networks are CW-complexes===

A ''<math>CW</math> complex'' , denoted as <math>X_c</math>, is a special [[../Bijective/|type]] of [[../CoIntersections/|topological]] space (<math>X</math>) which is the ''[[../ModuleAlgebraic/|union]]'' of an expanding sequence of subspaces <math>X^n</math>, such that, inductively, the first member of this expansion sequence is <math>X^0</math> -- a discrete set of points called the ''vertices''  of <math>X</math>, and <math>X^{n+1}</math> is the ''[[../Pushout/|pushout]]'' obtained from <math>X^n</math> by attaching disks <math>D^{n+1}</math> along "attaching maps" <math>j: S^n \rightarrow X^n</math>. Each resulting map <math>D^{n+1} \longrightarrow X</math> is called a ''cell'' . (The subscript "<math>c</math>" in <math>X_c</math>, stands for the fact that this (CW) type of topological space <math>X</math> is called ''cellular'' , or "made of cells"). The subspace <math>X^n</math> is called the "<math>n</math>-skeleton" of <math>X</math>.
Pushouts, expanding sequence and unions are here understood in the topological sense, with the compactly generated
topologies (''viz.''  p.71 in P. J. May, 1999 <ref name='MJP1999'/>).

'''Examples of a <math>CW</math> complex''' :

#A [[../Cod/|graph]] is a one--dimensional <math>CW</math> complex.

#''Spin networks''  are represented as graphs and they are therefore also one--dimensional <math>CW</math> complexes.
The transitions between ''spin networks''  lead to ''spin foams'' , and spin foams may be thus regarded
as a higher dimensional <math>CW</math> complex (of dimension <math>d \geq 2</math>).

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 <ref name='MJP1999'/> the '''Definition 0.1'''  presented here has advantages in proving
results, including generalized, or extended [[../Formula/|theorems]] in
[http://planetmath.org/?op=getobj&from=lec&id=73 Algebraic Topology],
(as for example in <ref name='MJP1999'/>).

==All Sources==
<ref name='MJP1999'/>
<ref name='CRFM1980'/>
<ref name='JJR1998'/>
<ref name='WH52'/>
<ref name='BF92'/>

==References==
<references>

</ref><ref name='MJP1999'>
May, J.P. 1999, ''A Concise Course in Algebraic Topology.'' , The University of Chicago Press: Chicago.

</ref><ref name='CRFM1980'>
C.R.F. Maunder. 1980, [http://planetmath.org/?op=getobj&from=books&id=181 Algebraic Topology.],
Dover Publications, Inc.: Mineola, New York.

</ref><ref name='JJR1998'>
Joseph J. Rothman. 1998,
[http://planetmath.org/?op=getobj&from=books&id=172 An Introduction to Algebraic Topology],
Springer-Verlag: Berlin

</ref><ref name='WH52'>
Werner Heisenberg. ''The Physical Principles of Quantum Theory'' . New York: Dover Publications, Inc.(1952), pp.39-47.

</ref><ref name='BF92'>
F. W. Byron, Jr. and R. W. Fuller. ''Mathematical Principles of Classical and Quantum Physics.'' , New York: Dover Publications, Inc. (1992).

</ref></references>

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