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operator algebra and complex representation theorems (Topic)

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

QAT theorems for quantum state spaces of spin networks and quantum spin foams based on $CW$-, $n$-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 $Z$ be a $CW$ complex that has the (three–dimensional) Quantum Spin `Foam' (QSF) as a subspace. Furthermore, let $f: Z \rightarrow QSS$ be a map so that $f \mid QSF = 1_{QSF}$, with QSS being an arbitrary, local quantum state space (which is not necessarily finite). There exists an $n$-connected $CW$ model (Z,QSF) for the pair (QSS,QSF) such that:

$f_*: \pi_i (Z) \rightarrow \pi_i (QST)$,

is an isomorphism for $i>n$ and it is a monomorphism for $i=n$. The $n$-connected $CW$ model is unique up to homotopy equivalence. (The $CW$ complex, $Z$, 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 $(QSS,QSF)$ of topological spaces defined as in Lemma 1, with QSF nonempty, there exist $n$-connected $CW$ models $f: (Z, QSF) \rightarrow (QSS, QSF)$ for all $n \geq 0$. Such models can be then selected to have the property that the $CW$ complex $Z$ is obtained from QSF by attaching cells of dimension $n>2$, and therefore $(Z,QSF)$ is $n$-connected. Following Lemma 01 one also has that the map: $f_* : \pi_i (Z) \rightarrow \pi_i (QSS)$ which is an isomorphism for $i>n$, and it is a monomorphism for $i=n$.

Note See also the definitions of (quantum) spin networks and spin foams.

Bibliography

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



"operator algebra and complex representation theorems" is owned by bci1.

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Also defines:  operator, quantum operator, observable, state function, quantum state space, QSS, local quantum state space, quantum spin network functor
Keywords:  operator, quantum operator, observable, state function, quantum state space, QSS, local quantum state space, quantum spin network functor

Cross-references: spin networks and spin foams, topological, section, homotopy, monomorphism, isomorphism, quantum spin foams, spin networks, theorems, QAT
There are 83 references to this object.

This is version 1 of operator algebra and complex representation theorems, born on 2009-03-03.
Object id is 567, canonical name is OperatorAlgebraAndComplexRepresentationTheorems2.
Accessed 2486 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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