NON–ABELIAN QUANTUM ALGEBRAIC TOPOLOGY: I. From Funda- mental Concepts to Physical Realizations in Quantum Gravity, Spin Networks, Superconductivity and Quantum Phase Transitions via Local-to-Global Con- structions v.2-x

Conceptual developments and a novel approach to Quantum theories are here 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 ‘machinery’ of Non–Abelian Algebraic Topology (NAAT), Category Theory (CT) and HigherDimensional 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)). New possibilities for the application of fundamental theorems from Algebraic Topology to physical processes such as Quantum Phase Transitions (e.g., as in superconductivity, colossal magnetoresistance and ferromagnetism), spin network fluctuations, parallel transport and quantum tunneling are also formulated. Among such fundamental theorems with potential physical applications in QAT is the Generalized van Kampen theorem (GvKT). Several other applications of AT fundamental theorems are suggested for the formulation of QST non–Abelian structural approximations by algebraic– topological, as well as logical, means. An extensive review and critical evaluation of published and archived articles, as well as standard texts, on QA, QFT, AQFT, TQFT, Supersymmetry/Supergravity (SG) and Gauge theories (GT) is summarized and provides essential concepts 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 ‘algebras’, 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. Under Construction, (in collaboration with J. F. Glazebrook* and R. Brown***)