An Introduction to Noncommutative Spaces and Their Geometries

Landi, Giovanni

ISBN 10: 3540635092 / ISBN 13: 9783540635093

Editore: Berlin/Heidelberg : Springer-Verlag, 1997

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Original boards, illustrated with numerous equations and diagrams, 8vo. Lecture Notes in Physics Monographs, 51.; Corners and edges worn, spine slightly discoloured, name in pen on title page. Codice articolo 342798-ZA29

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Titolo: An Introduction to Noncommutative Spaces and...

Casa editrice: Berlin/Heidelberg : Springer-Verlag

Data di pubblicazione: 1997

Legatura: Brossura

Condizione: Fair

Descrizione articolo

Riassunto:

These lecture notes are an introduction to several ideas and applications of noncommutative geometry. It starts with a not necessarily commutative but associative algebra which is thought of as the algebra of functions on some 'virtual noncommutative space'. Attention is switched from spaces, which in general do not even exist, to algebras of functions. In these notes, particular emphasis is put on seeing noncommutative spaces as concrete spaces, namely as a collection of points with a topology. The necessary mathematical tools are presented in a systematic and accessible way and include among other things, C'*-algebras, module theory and K-theory, spectral calculus, forms and connection theory. Application to Yang--Mills, fermionic, and gravity models are described. Also the spectral action and the related invariance under automorphism of the algebra is illustrated. Some recent work on noncommutative lattices is presented. These lattices arose as topologically nontrivial approximations to 'contuinuum' topological spaces. They have been used to construct quantum-mechanical and field-theory models, alternative models to lattice gauge theory, with nontrivial topological content. This book will be essential to physicists and mathematicians with an interest in noncommutative geometry and its uses in physics.

Contenuti:

Noncommutative Spaces and Algebras of Functions.- Projective Systems of Noncommutative Lattices.- Modules as Bundles.- A Few Elements of K-Theory.- The Spectral Calculus.- Noncommutative Differential Forms.- Connections on Modules.- Field Theories on Modules.- Gravity Models.- Quantum Mechanical Models on Noncommutative Lattices.