PlanetPhysics/Non Abelian Quantum Algebraic Topology 2

Non-Abelian Quantum Algebraic Topology (NAQAT)

This is a new contributed topic (under construction).

Quantum Algebraic Topology is the area of [[../NonNewtonian2/|theoretical physics]] and [[../NonNewtonian2/|physical mathematics]] concerned with the applications of [[../CubicalHigherHomotopyGroupoid/|algebraic topology]] methods, results and constructions (including its extensions to [[../TrivialGroupoid/|category theory]], [[../GrothendieckTopos/|topos]] Theory and [[../HigherDimensionalAlgebra2/|higher dimensional algebra]]) to fundamental quantum physics problems, such as the [[../CategoricalGroupRepresentation/|representations]] of Quantum spacetimes and Quantum State Spaces in [[../LQG2/|quantum gravity]], in arbitrary [[../CosmologicalConstant2/|reference frames]]. Non--Abelian gauge [[../CosmologicalConstant2/|field]] theories can also be formalized or presented in the [[../QATs/|QAT]] framework.

Perhaps the neighbor areas with which QAT overlaps significantly are: [[../CoIntersections/|algebraic]] [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theories]] (AQFT[[../MathematicalFoundationsOfQuantumTheories/|)/local quantum physics]] (LQP), Axiomatic QFT, Lattice [[../HotFusion/|QFT]] ([[../LQG2/|LQFT]]) and [[../Supersymmetry/|supersymmetry/]]. One can also claim overlap with various [[../CoIntersections/|topological]] Field Theories (TFT), or Topological Quantum Field Theories (TQFT), [[../ThinEquivalence/|homotopy]] QFT ([[../NonNewtonian2/|HQFT]]), Dilaton, and Lattice Quantum Gravity (respectively, DQG and [[../LQG2/|LQG]]) theories.

Applications of the Van Kampen Theorem to Crossed Complexes. Representations of Quantum Space-Time in terms of Quantum Crossed Complexes over a Quantum Groupoid.

There are several possible applications of the [[../SingularComplexOfASpace/|generalized Van Kampen theorem]] in the development of physical representations of a quantized [[../SR/|space-time]] `geometry' For example, a possible application of the generalized van Kampen theorem is the construction of the initial, quantized space-time as the unique colimit of quantum causal sets (posets) in terms of the nerve of an open [[../CubicalHigherHomotopyGroupoid/|covering]] $N^{‴}{U}^{‴}$ of the topological space $X$ that would be isomorphic to a $k$-simplex $K$ underlying $X$. The corresponding,[[../AbelianCategory3/|noncommutative]] algebra $\mathrm{\Omega }$ associated with the finitary $T_0$-poset $P(S)$ is the Rota algebra $\mathrm{\Omega }$, and the quantum topology $T_0$ is defined by the partial ordering arrows for regions that can overlap, or superpose, coherently (in the quantum sense) with each other. When the poset $P(S)$ contains $2N$ points we write this as $P_{2N}(S)$. The unique (up to an [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]]) $P(S)$ in the colimit , $\underleftarrow{\lim }{P}_{N}X$, recovers a space homeomorphic to $X$. Other non-Abelian results derived from the generalized van Kampen theorem were discussed by Brown, Hardie, Kamps and Porter, and also by Brown, Higgins and Sivera.

Local--to--Global (LG) Construction Principles consistent with Quantum Axiomatics

A novel approach to [[../SUSY2/|QST]] construction in AQFT may involve the use of fundamental [[../Formula/|theorems]] of algebraic topology generalised from topological spaces to spaces with structure, such as a filtration, or as an $n$-cube of spaces. In this [[../Cod/|category]] are the generalized, \htmladdnormallink{higher homotopy {http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} Seifert-van Kampen theorems (HHSvKT)} of Algebraic Topology with novel and unique non-Abelian applications. Such theorems have allowed some new calculations of homotopy [[../Bijective/|types]] of topological spaces. They have also allowed new proofs and generalisations of the classical [[../ModuleAlgebraic/|relative Hurewicz theorem]] by R. Brown and coworkers. One may find links of such results to the expected \emph [`non-commutative']{http://planetphysics.us/encyclopedia/AbelianCategory3.html} geometrical structure of quantized space--time.

See also the Exposition on NAQAT at: http://aux.planetphysics.org/files/lec/61/ANAQAT20e[[../LebesgueMeasure/|.pdf]]