PlanetPhysics/CW Complex Representation Theorems

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CW-complex representation theorems in quantum algebraic topology[edit | edit source]

\htmladdnormallink{QAT {http://planetphysics.us/encyclopedia/QAT.html} theorems for quantum state spaces of spin networks and quantum spin foams based on , -connected models and fundamental theorems.}

Let us consider first a lemma in order to facilitate the proof of the following theorem concerning spin networks and quantum spin foams.

Lemma Let be a complex that has the (three--dimensional) Quantum Spin `Foam' (QSF) as a subspace. Furthermore, let be a map so that Failed to parse (syntax error): {\displaystyle f \mid QSF = 1_{QSF'' } , with QSS being an arbitrary, local quantum state space (which is not necessarily finite). There exists an -connected model (Z,QSF) for the pair (QSS,QSF) such that}:

,

is an isomorphism for and it is a monomorphism for . The -connected model is unique up to homotopy equivalence. (The complex, , considered here is a homotopic `hybrid' between QSF and QSS).

Theorem 2. (Baianu, Brown and Glazebrook, 2007: In Section 9 of a recent NAQAT preprint). For every pair of topological spaces defined as in Lemma 1 , with QSF nonempty, there exist -connected models for all . Such models can be then selected to have the property that the complex is obtained from QSF by attaching cells of dimension , and therefore is -connected. Following Lemma 01 one also has that the map: which is an isomorphism for , and it is a monomorphism for .

Note See also the definitions of (quantum) \htmladdnormallink{spin networks and spin foams {http://planetphysics.us/encyclopedia/SpinNetworksAndSpinFoams.html}.}