PlanetPhysics/Theorem on CW Complex Approximation of Quantum State Spaces in QAT

\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html} 1.}

Let $$[QF_j]_{j=1,...,n}$$ be a complete sequence of commuting quantum spin `foams' (QSFs) in an arbitrary quantum state space (QSS), and let $$(QF_j,QSS_j)$$ be the corresponding sequence of pair subspaces of QST. If $$Z_j$$ is a sequence of CW-complexes such that for any $$j$$, $$QF_j \subset Z_j$$, then there exists a sequence of $$n$$-connected models $$(QF_j,Z_j)$$ of $$(QF_j,QSS_j)$$ and a sequence of induced isomorphisms $${f_*}^j : \pi_i (Z_j)\rightarrow \pi_i (QSS_j)$$ for $$i>n$$, together with a sequence of induced monomorphisms for $$i=n$$.

There exist weak homotopy equivalences between each $$Z_j$$ and $$QSS_j$$ spaces in such a sequence. Therefore, there exists a $$CW$$--complex approximation of QSS defined by the sequence $$[Z_j]_{j=1,...,n}$$ of CW-complexes with dimension $$n \geq 2$$. This $$CW$$--approximation is unique up to regular homotopy equivalence.

Corollary 2.

The $$n$$-connected models $$(QF_j,Z_j)$$ of $$(QF_j,QSS_j)$$ form the Model category of Quantum Spin Foams $$(QF_j)$$, whose \htmladdnormallink{morphisms {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are maps $$h_{jk}: Z_j \rightarrow Z_k$$ such that $$h_{jk}\mid QF_j = g: (QSS_j, QF_j) \rightarrow (QSS_k,QF_k)$$, and also such that the following diagram is commutative:} \\

\begin{CD} Z_j @> f_j >> QSS_j \\ @V h_{jk} VV  @VV g V \\ Z_k @ > f_k >> QSS_k \end{CD}

\\ Furthermore, the maps $$h_{jk $$ are unique up to the homotopy rel $$QF_j$$, and also rel $$QF_k$$}.

{Theorem 1} complements other data presented in the parent entry on QAT.