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%%% Primary Title: spin networks and spin foams
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\begin{document}
\begin{definition} \emph{\htmladdnormallink{spin networks}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}} are one-dimensional $CW$ complexes consisting of quantum \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} states of \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html}, defined by elements of Pauli \htmladdnormallink{matrices}{http://planetphysics.us/encyclopedia/Matrix.html} represented as vertices of a directed \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Cod.html} or network, and with the edges of the network representing the connections, or links, between such quantum spin states.
\end{definition}
\begin{remark} \textbf{On current \emph{formal} definitions of spin networks.}
For quantum \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} with known standard symmetry formal definitions of spin networks have also been reported in terms
of \htmladdnormallink{symmetry group}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html} \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}. An example of such a formal definition in terms of \htmladdnormallink{Lie group}{http://planetphysics.us/encyclopedia/BilinearMap.html} representations on \htmladdnormallink{Hilbert spaces}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} of quantum states and \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} is provided next.
\end{remark}
\begin{definition}
\emph{Spin networks} are formally defined here for quantum systems with `standard'* \htmladdnormallink{quantum symmetry}{http://planetphysics.us/encyclopedia/HilbertBundle.html} in terms of Lie group ($G_L$) \htmladdnormallink{irreducible representations}{http://planetphysics.us/encyclopedia/GroupRepresentation.html} on complex Hilbert spaces $\H$ of quantum states and \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} operators; such representations are precisely defined by special \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \htmladdnormallink{homomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} as follows.
Consider $Re$ as a Lie group $G_L$, and also consider the complex Hilbert space $\H$ to be $B[\H]$, the group of bounded \htmladdnormallink{linear operators}{http://planetphysics.us/encyclopedia/Commutator.html} of $\H$ which have a bounded inverse, and more specifically to be $L^2(Re)$.
Then, one defines the \emph{$G_L$-representation} as the group homomorphism $\rho: Re \to B[L^2(Re)]$ with
$\rho(r): \left\{f(x)\right\} \mapsto f(r^{-1}x)$, where $r \in Re$ and $f(x) \in L^2(Re)$.
*The word `standard' is employed here with the meaning of the \htmladdnormallink{Standard Model of physics}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} (\htmladdnormallink{SUSY}{http://planetphysics.us/encyclopedia/SUSY2.html}) which does \emph{not}
include either \emph{\htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html}} or its \htmladdnormallink{extended quantum symmetries}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html}.
\end{definition}
\begin{definition} \emph{\htmladdnormallink{spin foams}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}} are \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} $CW$ complexes representing two local
spin networks as described in \textbf{Definition 0.1} with quantum transitions between them; \emph{spin foams} are sometimes also represented by \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of spin networks considered as (small) \htmladdnormallink{categories}{http://planetphysics.us/encyclopedia/Cod.html} (\emph{viz.} Baez and Dollan,1998a,b; \cite{BAJ-DJ98a, BAJ-DJ98b}).
\end{definition}
For the sake of completeness, let us recall here the following
\begin{definition} a \emph{$CW$ complex}, $X_c$ is a \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space which is the \htmladdnormallink{union}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} of an expanding
sequence of subspaces $X^n$ such that, inductively, $X^0$ is a discrete set of points called
vertices and $X^{n+1}$ is the \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 subspace $X^n$ is called the ``$n$-skeleton'' of X.
\end{definition}
\textbf{An Example} of a $CW$ complex is a graph or `network' regarded as a one-dimensional $CW$ complex.
\textbf{Remark:}
Such `purely' topological definitions seem to miss much of the associated \htmladdnormallink{quantum operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} \htmladdnormallink{algebraic structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}
that are essential to the mathematical foundation of \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}; note however the first related entry
that addresses this important, \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} question.
\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{Quantum Gravity} \htmladdnormallink{spacetimes}{http://planetphysics.us/encyclopedia/SR.html}. The {\em spin observable}-- which is fundamental in quantum theories-- has no corresponding concept in \htmladdnormallink{classical mechanics}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html}. (However, classical \emph{momenta} (both linear and angular) have corresponding \htmladdnormallink{quantum observable}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} operators 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 might expect 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/CosmologicalConstant.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'. This \emph{intrinsic}, spin observable, can also be understood as an \emph\emph{internal symmetry} of quantum particles, which in many cases can be understood in terms of `internal' symmetry group representations, such as the Dirac or Pauli matrices that are currently employed in quantum mechanics, \htmladdnormallink{quantum electrodynamics}{http://planetphysics.us/encyclopedia/QED.html}, \htmladdnormallink{QCD}{http://planetphysics.us/encyclopedia/HotFusion.html} and QFT. There are thus \emph{\htmladdnormallink{fermion}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html}} (quantum) symmetries, quantum statistics, etc, for quantum particles with half-integer spin values (for massive particles such as electrons, protons,neutrons, \htmladdnormallink{quarks}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html}, nuclei with an \emph{odd} number of \htmladdnormallink{nucleons}{http://planetphysics.us/encyclopedia/Pions.html}) and \emph{\htmladdnormallink{boson}{http://planetphysics.us/encyclopedia/BoseEinsteinStatistics.html}} (quantum) symmetries, statistics, etc., for quantum particles with integer spin values, such as $0, 1, 2, ..., n$, {where $n$ is usually thought to be less than $3$, for field quanta such as photons, gravitons, gluons, hypothetical \htmladdnormallink{Higgs bosons}{http://planetphysics.us/encyclopedia/Neutralinos.html}, etc).
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 if the spin value is non-zero (i.e., generally, an integer, or half-integer) thus giving rise to ``spin networks'', which can be mathematically represented as in \textbf{Defintion 0.1} 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'.
As a practical (and thus `intuitive', pictorial) example, the \htmladdnormallink{detection}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html} of all
\htmladdnormallink{MRI (2D-FT) images}{http://planetphysics.us/encyclopedia/TwoDimensionalFourierTransforms.html} employed in
\htmladdnormallink{clinical medicine and biomedical research}{http://www.wpi.edu/Pubs/ETD/Available/etd-081707-080430/unrestricted/dbennett.pdf}, as well as all (multi-) Nuclear Magnetic \htmladdnormallink{resonance}{http://planetphysics.us/encyclopedia/QualityFactorOfAResonantCircuit.html} (\htmladdnormallink{NMR}{http://planetphysics.us/encyclopedia/SpectralImaging.html}) spectra employed in physical, chemical, biophyisical/biochemical/biomedical, polymer and agricultural research involves quantum transitions between spin networks or spin foams.
\begin{thebibliography}{99}
\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).
\bibitem{BAJ-DJ98a}
Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in \emph{Advances in Mathematics}, \textbf{135}: 145-206.
\bibitem{BAJ-DJ98b}
Baez, J. \& Dolan, J., 1998b, \emph{``Categorification'', Higher Category Theory, Contemporary Mathematics},
\textbf{230}, Providence: \emph{AMS}, 1-36.
\bibitem{BAJ-DJ2k1}
Baez, J. \& 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}
