CW-complex of spin networks (CWSN)

Definition 0.1. A CW complex, denoted as X_c, is a special type of topological space (X) which is the union of an expanding sequence of subspaces Xⁿ, such that, inductively, the first member of this expansion sequence is X⁰ – a discrete set of points called the vertices of X, 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 subscript “c” in X_c, stands for the fact that this (CW) type of topological space X is called cellular, or “made of cells”). The subspace Xⁿ is called the “n-skeleton” of X. 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 CW complex:

A graph is a one–dimensional CW complex.
spin networks are represented as graphs and they are therefore also one–dimensional CW complexes. The transitions between spin networks lead to spin foams, and spin foams may be thus regarded as a higher dimensional CW complex (of dimension d ≥ 2).

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]).

References

[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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