CW-complex of spin networks (CWSN)
Definition 0.1
A
complex, denoted as

, is a special
type
of
topological
space (

) which is the
union of an expanding sequence of subspaces

, such that, inductively, the first member of this expansion sequence is

- a discrete set of points called the
vertices of

, and

is the
pushout obtained from

by attaching disks

along “attaching maps”

. Each resulting map

is called a
cell. (The subscript “

” in

, stands for the fact that this (CW) type of topological space

is called
cellular, or “made of cells”). The subspace

is called the “

-skeleton” of

.
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 [
1]).
Examples of a
complex:
- A graph
is a one-dimensional
complex.
- spin networks are represented as graphs and they are therefore also one-dimensional
complexes.
The transitions between spin networks lead to spin foams, and spin foams may be thus regarded
as a higher dimensional
complex (of dimension
).
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 that might be expected 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'. 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 [4])). 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'.
Remark 0.1
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 [
1] the
Definition 0.1 presented here has advantages in proving
results, including generalized, or extended
theorems
in
Algebraic Topology,
(as for example in [
1]).
- 1
-
May, J.P. 1999, A Concise Course in Algebraic Topology., The University of Chicago Press: Chicago.
- 2
-
C.R.F. Maunder. 1980, Algebraic Topology.,
Dover Publications, Inc.: Mineola, New York.
- 3
-
Joseph J. Rothman. 1998,
An Introduction to Algebraic Topology,
Springer-Verlag: Berlin
- 4
-
Werner Heisenberg. The Physical Principles of Quantum Theory. New York: Dover Publications, Inc.(1952), pp.39-47.
- 5
-
F. W. Byron, Jr. and R. W. Fuller. Mathematical Principles of Classical and Quantum Physics., New York: Dover Publications, Inc. (1992).
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