PlanetPhysics/Operator Algebra and Complex Representation Theorems 2

CW-complex representation theorems in quantum operator algebra and quantum algebraic topologyEdit

\htmladdnormallink{QAT {} 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 ref. [1]. 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 {}.}

All SourcesEdit



  1. 1.0 1.1 I. C. Baianu, J. F. Glazebrook and R. Brown.2008.Non-Abelian Quantum Algebraic Topology, pp.123 Preprint.