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fundamental groupoid functors in quantum theories (Topic)

Fundamental Groupoid Functors in Quantum Theories

The natural setting for the definition of a quantum fundamental groupoid $F_{\mathcal Q}$ is in one of the functor categories– that of fundamental groupoid functors, $F_{\mathcal G}$, and their natural transformations defined in the context of quantum categories of quantum spaces ${\mathcal Q}$ represented by Hilbert space bundles or `rigged' Hilbert (or Frechét) spaces ${\H }_B$.

Let us briefly recall the description of quantum fundamental groupoids in a quantum functor category, ${\mathcal Q}_F$:

Definition 0.1   The quantum fundamental groupoid, QFG is defined by a functor $F_{\mathcal Q}: \H _B \to {\mathcal Q}_G$, where ${\mathcal Q}_G$ is the category of quantum groupoids and their homomorphisms.

Fundamental Groupoid Functors

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, $F_{\mathcal G}: \textbf{Top} \to \mathcal G_2$, where Top is the category of topological spaces and $\mathcal G_2$ is the groupoid category.

Specific Example of QFG

One can provide a physically relevant example of QFG as spin foams, or functors of spin networks; more precise the spin foams were defined as functors between spin network categories that realize dynamic transformations on the spin space. Thus, because spin networks (or graphs) are specialized one-dimensional CW-complexes whose cells are linked quantum spin states their quantum fundamental groupoid is defined as a representation of CW-complexes on `rigged' Hilbert spaces, that are called Frechét nuclear spaces.



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See Also: fundamental quantum groupoid

Also defines:  fundamental groupoid functor
Keywords:  fundamental groupoid functor, quantum fundamental groupoid, spin networks, spin foams, functors and natural transformations, dynamics on spin spaces

Cross-references: rigged Hilbert spaces, representation, graphs, spin, dynamic, spin networks, spin foams, topological, general definition, homomorphisms, quantum groupoids, category, functor category, Hilbert space bundles, quantum categories, functor, quantum fundamental groupoid
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This is version 4 of fundamental groupoid functors in quantum theories, born on 2009-02-04, modified 2009-02-13.
Object id is 499, canonical name is QuantumFundamentalGroupoid.
Accessed 2009 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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