Definition 0.1   spin networks are one-dimensional 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.

Definition 0.2   Spin networks are formally defined here for quantum systems with `standard'* quantum symmetry in terms of Lie group () irreducible representations on complex Hilbert spaces of quantum states and observable operators; such representations are precisely defined by special group homomorphisms as follows. Consider as a Lie group , and also consider the complex Hilbert space to be , the group of bounded linear operators of which have a bounded inverse, and more specifically to be . Then, one defines the -representation as the group homomorphism with , where and .

*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 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 complex, is a topological space which is the union of an expanding sequence of subspaces such that, inductively, is a discrete set of points called vertices and is the pushout obtained from by attaching disks along “attaching maps” . Each resulting map is called a cell. The subspace is called the “-skeleton” of X.

An Example of a complex is a graph or `network' regarded as a one-dimensional 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 , where is usually thought to be less than , 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.

Bibliography

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.