Definition 0.1. spin networks are one-dimensional CW complexes consisting of quantum
spin states of particles, defined by elements of Pauli matrices represented as vertices of a
directed graph or network, and with the edges of the network representing the connections,
or links, between such quantum spin states.
Remark 0.1. On current formal definitions of spin networks. For quantum systems
with known standard symmetry formal definitions of spin networks have also been reported
in terms of symmetry group representations. An example of such a formal definition in terms
of Lie group representations on Hilbert spaces of quantum states and operators is provided
next.
Definition 0.2. Spin networks are formally defined here for quantum systems with
‘standard’* quantum symmetry in terms of Lie group (GL) irreducible representations on
complex Hilbert spaces of quantum states and observable operators; such representations
are precisely defined by special group homomorphisms as follows. Consider Re as a Lie
group GL, and also consider the complex Hilbert space to be B[], the group of bounded
linear operators of which have a bounded inverse, and more specifically to be L2(Re). Then,
one defines the GL-representation as the group homomorphism ρ : Re → B[L2(Re)] with
ρ(r) : 
f(r−1x), where r ∈ Re and f(x) ∈ L2(Re).
*The word ‘standard’ is employed here with the meaning of the Standard Model of physics
(SUSY) which does not include either quantum gravity or its extended quantum symmetries.
Definition 0.3. spin foams are 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 functors of spin networks considered as (small) categories
(viz. Baez and Dollan,1998a,b; [3, 4]).
For the sake of completeness, let us recall here the following
Definition 0.4. a CW complex, Xc is a topological space which is the union of an expanding
sequence of subspaces Xn such that, inductively, X0 is a discrete set of points called vertices
and Xn+1 is the pushout obtained from Xn by attaching disks Dn+1 along “attaching maps”
j : Sn → Xn. Each resulting map Dn+1→X is called a cell. The subspace Xn 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 quantum
operator algebraic structures that are essential to the mathematical foundation of quantum
theories; note however the first related entry that addresses this important, algebraic
question.
Note. The concepts of spin networks and spin foams were recently developed in the context of
mathematical physics as part of the more general effort of attempting to formulate mathematically
a concept of quantum state space which is also applicable, or relates to Quantum Gravity
spacetimes. The spin observable– which is fundamental in quantum theories– has no corresponding
concept in classical mechanics. (However, classical momenta (both linear and angular) have
corresponding quantum observable operators that are quite different in form, with their
eigenvalues taking on different sets of values in quantum mechanics than the ones might
expect from classical mechanics for the ‘corresponding’ classical observables); the spin is
an intrinsic observable of all massive quantum ‘particles’, such as electrons, protons,
neutrons, atoms, as well as of all field quanta, such as photons, gravitons, gluons, and
so on; furthermore, every quantum ‘particle’ has also associated with it a de Broglie
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, quantum electrodynamics, QCD and QFT. There are thus fermion (quantum)
symmetries, quantum statistics, etc, for quantum particles with half-integer spin values
(for massive particles such as electrons, protons,neutrons, quarks, nuclei with an odd
number of nucleons) and 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 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 magnetic field as in the
famous Stern-Gerlach experiment, (although the applied field may also be electric or
gravitational, (see for example [1])). 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 detection of all MRI (2D-FT) images
employed in clinical medicine and biomedical research, as well as all (multi-) Nuclear Magnetic
resonance (NMR) spectra employed in physical, chemical, biophyisical/biochemical/biomedical,
polymer and agricultural research involves quantum transitions between spin networks or spin
foams.
References
[1] Werner Heisenberg. The Physical Principles of Quantum Theory. New York: Dover
Publications, Inc.(1952), pp.39-47.
[2] F. W. Byron, Jr. and R. W. Fuller. Mathematical Principles of Classical and
Quantum Physics., New York: Dover Publications, Inc. (1992).
[3] Baez, J. & Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the
Algebra of Opetopes, in Advances in Mathematics, 135: 145-206.
[4] Baez, J. & Dolan, J., 1998b, “Categorification”, Higher Category Theory,
Contemporary Mathematics, 230, Providence: AMS, 1-36.
[5] Baez, J. & Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in Mathematics
Unlimited – 2001 and Beyond, Berlin: Springer, pp. 29–50.
