Comments:

2008 update, preprint, 135-page monograph

Description:

Conceptual developments and a novel approach to Quantum theories are proposed starting from existing Quantum Group Algebra (QGA), Algebraic Quantum Field Theories (AQFT), standard and effective Quantum Field Theories (QFT), as well as the refined tools of Non--Abelian Algebraic Topology (NAAT), Category Theory (CT) and Higher-Dimensional Algebra (HDA). The open question of building valid QST representations in Quantum Gravity (QG) is approached as a Local-to-Global (LG) mathematical construction problem in Quantum Algebraic Topology (QAT). QST representations are here proposed for quantum systems with either finite (Quantum Mechanics and Quantum Automata) or infinite degrees of freedom (Quantum Field Theories (QFT)). Potential applications of fundamental theorems from Algebraic Topology to physical processes are suggested, such as: Quantum Phase Transitions (e.g., as in superconductivity, colossal magnetoresistance and ferromagnetism), spin network fluctuations, parallel transport and quantum tunneling. Among the fundamental theorems with potential physical applications in NAQAT is the Generalized van Kampen theorem (GvKT). A critical review and evaluation of published and archived articles, as well as standard texts, on QA, QFT, AQFT, TQFT, Supersymmetry/Supergravity (SG) and Gauge theories (GT) is also concisely presented; this evaluatiom leads to essential concepts that are relevant to the development of improved mathematical representations of Quantum Space-Time (QST) and Quantum State Spaces (QSS). Among such key concepts are: Quantum Group Algebras (QGAs)/Groupoids, Hopf and C*-algebras, Lie-groups, Lie-algberas,Lie Algebroids, Crossed Complexes over Groupoids, and CW-complexes of Spin Networks and Quantum Spin Foams. These important concepts are presented in a sequence tailored to aid the development of a non-Abelian, algebraic topological framework for QG theories of intense gravitational fields in curved, fluctuating QST. The concepts presented in this paper are intended to fill gaps between Quantum Field theories and General Relativity theory, thus leading towards generally relativistic Quantum Gravity in a non-Abelian framework of fluctuating, Quantum Space-Time. Related links: http://www.bangor.ac.uk/~mas010/nonab-a-t.html http://arxiv.org/abs/math.AT/0407275v1

Rights:

Open access http://www.bangor.ac.uk/~mas010/nonab-a-t.html http://arxiv.org/abs/math.AT/0407275v1 http://www.ag.uiuc.edu/~fs401/Quantum%20Algebraic%20Topology.pdf

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ANAQAT20b.pdf

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