PlanetPhysics/Spin Networks and Spin Foams

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 [[../SimplicialCWComplex/|spin networks]] are one-dimensional $CW$ complexes consisting of quantum [[../QuarkAntiquarkPair/|spin]] states of [[../Particle/|particles]], defined by elements of Pauli [[../Matrix/|matrices]] represented as vertices of a directed [[../Cod/|graph]] or network, and with the edges of the network representing the connections, or links, between such quantum spin states.

On current formal  definitions of spin networks.

For quantum [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] with known standard symmetry formal definitions of spin networks have also been reported in terms of [[../TopologicalOrder2/|symmetry group]] [[../CategoricalGroupRepresentation/|representations]]. An example of such a formal definition in terms of [[../BilinearMap/|Lie group]] representations on [[../NormInducedByInnerProduct/|Hilbert spaces]] of quantum states and [[../QuantumOperatorAlgebra4/|operators]] is provided next.

Spin networks are formally defined here for quantum systems with `standard'* [[../HilbertBundle/|quantum symmetry]] in terms of Lie group ($G_L$) [[../GroupRepresentation/|irreducible representations]] on complex Hilbert spaces $ℍ$ of quantum states and [[../QuantumSpinNetworkFunctor2/|observable]] operators; such representations are precisely defined by special [[../TrivialGroupoid/|group]] [[../TrivialGroupoid/|homomorphisms]] 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 [[../Commutator/|linear operators]] of $ℍ$ which have a bounded inverse, and more specifically to be $L^2(Re)$. Then, one defines the $G_L$-representation as the group homomorphism $\rho :Re\to B[L^2(Re)]$ with $\rho (r):\{f(x)\}\mapsto f(r^{-1}x)$, where $r\in Re$ and $f(x)\in L^2(Re)$.

• The word 'standard' is employed here with the meaning of the [[../QuarkAntiquarkPair/|Standard Model of physics]] ([[../SUSY2/|SUSY]]) which does not

include either [[../LQG2/|quantum gravity]] or its [[../TopologicalOrder2/|extended quantum symmetries]].

[[../SimplicialCWComplex/|spin foams]] are [[../CoriolisEffect/|two-dimensional]] $CW$ complexes representing two local

spin networks as described in Definition 0.1 with quantum transitions between them; spin foams are sometimes also represented by [[../TrivialGroupoid/|functors]] of spin networks considered as (small) [[../Cod/|categories]] (viz. Baez and Dollan,1998a,b; ).

For the sake of completeness, let us recall here the following

a $CW$ complex , $X_c$ is a [[../CoIntersections/|topological]] space which is the [[../ModuleAlgebraic/|union]] 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 [[../Pushout/|pushout]] obtained from $X^n$ by attaching disks $D^{n+1}$ along "attaching maps" $j:S^n\to X^n$. Each resulting map $D^{n+1}⟶X$ is called a cell . The subspace $X^n$ is called the "$n$-skeleton" of X.

An Example of a $CW$ complex is a graph or `network' regarded as a one-dimensional $CW$ complex.

Remark: Such `purely' topological definitions seem to miss much of the associated [[../QuantumOperatorAlgebra5/|quantum operator]] [[../TrivialGroupoid/|algebraic structures]] that are essential to the mathematical foundation of [[../QuantumOperatorAlgebra5/|quantum theories]]; note however the first related entry that addresses this important, [[../CoIntersections/|algebraic]] question.

Note. 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 Quantum Gravity [[../SR/|spacetimes]]. The spin observable -- which is fundamental in quantum theories-- has no corresponding concept in [[../NewtonianMechanics/|classical mechanics]]. (However, classical momenta (both linear and angular) have corresponding [[../QuantumOperatorAlgebra5/|quantum observable]] operators that are quite different in form, with their eigenvalues taking on different sets of values in [[../QuantumParadox/|quantum mechanics]] than the ones might expect 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 [[../CosmologicalConstant/|wave]], so that it cannot be realized, or `pictured', as any kind of classical `body'. This intrinsic , spin observable, can also be understood as an 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, [[../QED/|quantum electrodynamics]], [[../HotFusion/|QCD]] and QFT. There are thus [[../QuarkAntiquarkPair/|fermion]] (quantum) symmetries, quantum statistics, etc, for quantum particles with half-integer spin values (for massive particles such as electrons, protons,neutrons, [[../ExtendedQuantumSymmetries/|quarks]], nuclei with an odd number of [[../Pions/|nucleons]]) and [[../BoseEinsteinStatistics/|boson]] (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 [[../Neutralinos/|Higgs bosons]], 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 [[../NeutrinoRestMass/|magnetic field]] as in the famous Stern-Gerlach experiment, (although the applied field may also be electric or gravitational, (see for example )). 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 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 [[../MolecularOrbitals/|detection]] of all [[../TwoDimensionalFourierTransforms/|MRI (2D-FT) images]] employed in clinical medicine and biomedical research, as well as all (multi-) Nuclear Magnetic [[../QualityFactorOfAResonantCircuit/|resonance]] ([[../SpectralImaging/|NMR]]) spectra employed in physical, chemical, biophyisical/biochemical/biomedical, polymer and agricultural research involves quantum transitions between spin networks or spin foams.