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439 pages, 8.7 Mb, June 16, 2009

Abstract:

Foundations of Quantum Algebraic Topology and Quantum Logic. See also Algebraic Quantum Field Theory (AQFT). 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 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)). AMS MSC: 55U35 (Algebraic topology :: Applied homological algebra and category theory :: Abstract and axiomatic homotopy theory) 81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods) 22A22 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Topological groupoids ) 46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras) 18B40 (Category theory; homological algebra :: Special categories :: Groupoids, semigroupoids, semigroups, groups ) 55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory) 22D25 (Topological groups, Lie groups :: Locally compact groups and their algebras :: $C^*$-algebras and $W$*-algebras in relation to group representations) 43A25 (Abstract harmonic analysis :: Fourier and Fourier-Stieltjes transforms on locally compact abelian groups) 43A35 (Abstract harmonic analysis :: Positive definite functions on groups, semigroups, etc.) 46L87 (Functional analysis :: Selfadjoint operator algebras :: Noncommutative differential geometry) 20L05 (Group theory and generalizations :: Groupoids )

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Open access: http://en.wikipedia.org/w/index.php?title=Special:Book/download/&collection_id=8c73ead39040cbbd&writer=rl&return_to=User%3ABci2%2FBooks%2FQuantum+Algebraic+Topology Copyright@2009 Bci2 and bci1 Copyright Statement: Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version ;published by the Free Software Foundation, with no Invariant Sections, no Front-Cover Texts, and no Back- Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Edited by Bci2, with all contributors listed before the License statement.

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