CW-complex approximation of quantum state spaces in QAT

Theorem 1.

Let [QFj]j=1,,n be a complete sequence of commuting quantum spin ‘foams’ (QSFs) in an arbitrary quantum state spacePlanetmathPlanetmath (QSS) (, and let (QFj,QSSj) be the corresponding sequence of pair subspacesPlanetmathPlanetmath of QST. If Zj is a sequence of CW-complexesMathworldPlanetmath such that for any j , QFjZj, then there exists a sequence of n-connected models (QFj,Zj) of (QFj,QSSj) and a sequence of induced isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath f*j:πi(Zj)πi(QSSj) for i>n, together with a sequence of induced monomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmath for i=n.

Remark 0.1.

There exist weak homotopy equivalencesMathworldPlanetmathPlanetmath between each Zj and QSSj spaces in such a sequence. Therefore, there exists a CW–complex approximation of QSS defined by the sequence [Zj]j=1,,n of CW-complexes with dimensionPlanetmathPlanetmath n2. This CW–approximation is unique up to regularPlanetmathPlanetmath homotopy equivalence.

Corollary 2.

The n-connected models (QFj,Zj) of (QFj,QSSj) form the Model Category of Quantum Spin Foams ( (QFj), whose morphismsMathworldPlanetmath are maps hjk:ZjZk such that hjkQFj=g:(QSSj,QFj)(QSSk,QFk), and also such that the following diagram is commutativePlanetmathPlanetmathPlanetmath:

ZjfjQSSjhjkgZk@ >fkQSSk
Furthermore, the maps hjk are unique up to the homotopyMathworldPlanetmathPlanetmath rel QFj , and also rel QFk.

Remark 0.2.

Theorem 1 complements other data presented in the parent entry on QAT (

Title CW-complex approximation of quantum state spaces in QAT
Canonical name CWcomplexApproximationOfQuantumStateSpacesInQAT
Date of creation 2013-03-22 18:14:37
Last modified on 2013-03-22 18:14:37
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 29
Author bci1 (20947)
Entry type Theorem
Classification msc 81T25
Classification msc 81T05
Classification msc 81T10
Synonym quantum spin networks approximations by CW-complexes
Related topic ApproximationTheoremForAnArbitrarySpace
Related topic HomotopyEquivalence
Related topic QuantumAlgebraicTopology
Related topic ApproximationTheoremForAnArbitrarySpace
Related topic SpinNetworksAndSpinFoams
Related topic QuantumSpaceTimes
Defines CW-complex approximation of quantum state spaces in QAT