1 Non-Abelian Quantum Algebraic Topology (NAQAT)
This is a new contributed topic (under construction).
Quantum Algebraic Topology is the area of theoretical physics and physical mathematics concerned
with the applications of algebraic topology methods, results and constructions (including its
extensions to category theory, topos Theory and higher dimensional algebra) to fundamental
quantum physics problems, such as the representations of Quantum spacetimes and Quantum
State Spaces in quantum gravity, in arbitrary reference frames. Non–Abelian gauge field theories
can also be formalized or presented in the QAT framework.
Perhaps the neighbor areas with which QAT overlaps significantly are: algebraic quantum field
theories (AQFT)/local quantum physics (LQP), Axiomatic QFT, Lattice QFT (LQFT) and
supersymmetry/. One can also claim overlap with various topological Field Theories (TFT), or
Topological Quantum Field Theories (TQFT), homotopy QFT (HQFT), Dilaton, and Lattice
Quantum Gravity (respectively, DQG and LQG) theories.
1.1 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 generalized Van Kampen theorem in the development
of physical representations of a quantized 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
covering NU of the topological space X that would be isomorphic to a k-simplex K underlying X.
The corresponding,noncommutative algebra Ω associated with the finitary T0-poset P(S) is the
Rota algebra Ω, and the quantum topology T0 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 P2N(S). The unique (up
to an isomorphism) P(S) in the colimit, lim ←PNX, 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.
1.2 Local–to–Global (LG) Construction Principles consistent with Quantum Axiomatics
A novel approach to QST construction in AQFT may involve the use of fundamental 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 category are the generalized, higher homotopy Seifert-van
Kampen theorems (HHSvKT) of Algebraic Topology with novel and unique non-Abelian
applications. Such theorems have allowed some new calculations of homotopy types
of topological spaces. They have also allowed new proofs and generalisations of the
classical relative Hurewicz theorem by R. Brown and coworkers. One may find links
of such results to the expected non-commutative geometrical structure of quantized
space–time.
See also the Exposition on NAQAT at: ANAQAT20e.pdf
