-complex approximation of quantum state spaces in QAT
Theorem 1.
Let be a complete sequence of commuting quantum spin ‘foams’
(QSFs) in an arbitrary quantum state space (QSS) (http://planetmath.org/QuantumSpaceTimes), and let be the corresponding sequence of pair subspaces
of QST. If is a sequence of CW-complexes
such that for any
, , then there exists a sequence of -connected models of
and a sequence of induced isomorphisms
for , together with a sequence of induced monomorphisms
for .
Remark 0.1.
There exist weak homotopy equivalences between each and spaces
in such a sequence. Therefore, there exists a –complex approximation of QSS defined by the sequence
of CW-complexes with dimension
. This –approximation is
unique up to regular
homotopy equivalence.
Corollary 2.
The -connected models of form the Model Category of
Quantum Spin Foams (http://planetmath.org/SpinNetworksAndSpinFoams) , whose morphisms are maps such that , and also such that the following diagram is commutative
:
Furthermore, the maps are unique up to the homotopy rel , and also rel .
Remark 0.2.
Theorem 1 complements other data presented in the parent entry on QAT (http://planetmath.org/QuantumAlgebraicTopology).
Title | -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 -complexes |
Related topic | ApproximationTheoremForAnArbitrarySpace |
Related topic | HomotopyEquivalence |
Related topic | QuantumAlgebraicTopology |
Related topic | ApproximationTheoremForAnArbitrarySpace |
Related topic | SpinNetworksAndSpinFoams |
Related topic | QuantumSpaceTimes |
Defines | -complex approximation of quantum state spaces in QAT |