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2010 edition, version 20e, 87 pages, incl. Table of Contents

Abstract:

An outline of novel applications of non-Abelian Algebraic Topology to Local Quantum Physics and AQFT. 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 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); GvkT potential applications include the construction of 'global' QST representations in the physical context of relativistic quantum measurements or quantum gravity, and the precise embedding of smaller structures into larger ones in order to obtain exact solutions for the former. 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. The logical links between Quantum Operator Algebras and their corresponding,'dual' structure of the Quantum State Spaces are also investigated. 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', Quantization and Asymptotic Morphisms, Locally Topological Groupoids, Crossed Modules of Groups or Lie Double Groupoids, Lie Algebroids, Crossed Complexes over Groupoids, Holonomy and gauge transformation groupoids, Quantum Principal Bundles and Sheaves, 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.

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