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.
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 of the topological space
that would be isomorphic to a
-simplex
underlying
.
The corresponding,noncommutative algebra
associated
with the finitary
-poset
is the Rota algebra
, and the quantum topology
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
contains
points we write this as
. The unique (up to an isomorphism)
in the colimit,
, recovers a space homeomorphic to
. 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.
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 -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'http://planetphysics.org/encyclopedia/AbelianCategory3.html geometrical structure of
quantized space-time.
See also the Exposition on NAQAT at: http://aux.planetphysics.org/files/lec/61/ANAQAT20e.pdf
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