PlanetPhysics/Quantum Fundamental Groupoid

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Fundamental Groupoid Functors in Quantum Theories

The natural setting for the definition of a [[../QuantumFundamentalGroupoid4/|quantum fundamental groupoid]] $F_{\mathbb{Q}}$ is in one of the [[../TrivialGroupoid/|functor]] categories-- that of [[../FundamentalGroupoidFunctor/|fundamental groupoid functors]], Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}} , and their [[../NaturalTransformation/|natural transformations]] defined in the context of [[../QuantumCategories/|quantum categories]] of quantum spaces $\mathbb{Q}$ represented by [[../HilbertBundle/|Hilbert space bundles]] or `rigged' Hilbert (or Frech\'et) spaces ${ℍ}_B$.

Let us briefly recall the description of quantum fundamental groupoids in a quantum [[../TrivialGroupoid/|functor category]], ${\mathbb{Q}}_F$: The quantum fundamental groupoid , QFG is defined by a functor $F_{\mathbb{Q}}:{ℍ}_B\to {\mathbb{Q}}_G$, where ${\mathbb{Q}}_G$ is the [[../Cod/|category]] of [[../WeakHopfAlgebra/|quantum groupoids]] and their [[../TrivialGroupoid/|homomorphisms]].

Fundamental Groupoid Functors

Other related functor categories are those specified with the [[../PreciseIdea/|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 [[../CoIntersections/|topological]] spaces and Failed to parse (unknown function "\grp"): {\displaystyle \grp_2} is the [[../GroupoidCategory/|groupoid category]].

Specific Example of QFG

One can provide a physically relevant example of QFG as [[../SimplicialCWComplex/|spin foams]], or functors of [[../SimplicialCWComplex/|spin networks]]; more precise the spin foams were defined as functors between spin network categories that realize [[../NewtonianMechanics/|dynamic]] transformations on the [[../QuarkAntiquarkPair/|spin]] space. Thus, because spin networks (or [[../Cod/|graphs]]) are specialized one-dimensional CW-complexes whose cells are linked quantum spin states their quantum fundamental groupoid is defined as a [[../CategoricalGroupRepresentation/|representation]] of CW-complexes on `[[../I3/|rigged' Hilbert spaces]], that are called Frech\'et nuclear spaces .