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145 pages, Preprint, 2007

Abstract:

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 Higher-Dimensional Algebra (HDA). The open question of building valid QST representations in Quantum Gravity (QG) is approached as a \emph {Local-to-Global} (LG) mathematical construction problem in Quantum Algebraic Topology (QAT). QST representations are here proposed for quantum systems with either \emph{finite} (Quantum Mechanics and Quantum Automata) or \emph{infinite} degrees of freedom (Quantum Field Theories (QFT)). 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). These important concepts are presented in a sequence tailored to aid the development of a non-Abelian algebraic topology 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-Times.

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Copyright@2007-2009 I.C. Baianu, James F. Glazebrook and Ronald Brown http://www.bangor.ac.uk/~mas010/brownpr.html

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