PlanetPhysics/Quantum Fundamental Groupoid

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Fundamental Groupoid Functors in Quantum Theories

The natural setting for the definition of a quantum fundamental groupoid $F_{\mathbb {Q} }$ is in one of the functor categories-- that of fundamental groupoid functors, Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}} , and their natural transformations defined in the context of quantum categories of quantum spaces ${\mathbb {Q} }$ represented by Hilbert space bundles or `rigged' Hilbert (or Frech\'et) spaces ${\mathbb {H} }_{B}$.

Let us briefly recall the description of quantum fundamental groupoids in a quantum functor category, ${\mathbb {Q} }_{F}$: The quantum fundamental groupoid , QFG is defined by a functor $F_{\mathbb {Q} }:\mathbb {H} _{B}\to {\mathbb {Q} }_{G}$, where ${\mathbb {Q} }_{G}$ 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 , Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}: '''Top''' \to \grp_2} , where Top is the category of topological spaces and Failed to parse (unknown function "\grp"): {\displaystyle \grp_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 Frech\'et nuclear spaces .