An Introduction to Noncommutative Spaces and Their Geometries: 51 (Lecture Notes in Physics Monographs) - Softcover

Landi, Giovanni

Softcover

ISBN 10: 3662141094 ISBN 13: 9783662141090

Verlag: Springer, 2014

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Reseña del editor

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.

Reseña del editor

These notes arose from a series of introductory seminars on noncommutative geometry I gave at the University of Trieste in September 1995 during the X WorkshoponDi?erentialGeometricMethodsinClassicalMechanics.Itwas Beppe Marmo's suggestion that I wrote notes for the lectures. The notes are mainly an introduction to Connes' noncommutative geo- try. They could serve as a '?rst aid kit' before one ventures into the beautiful but bewildering landscape of Connes' theory. The main di?erence from other available introductions to Connes' work, is the emphasis on noncommutative spaces seen as concrete spaces. Important examples of noncommutative spaces are provided by nonc- mutative lattices. The latter are the subject of intense work I am doing in collaboration with A.P. Balachandran, Giuseppe Bimonte, Elisa Ercolessi, Fedele Lizzi, Gianni Sparano and Paulo Teotonio-Sobrinho. These notes are also meant to be an introduction to this research. There is still a lot of work in progress and by no means can these notes be considered as a review of - erything we have achieved so far. Rather, I hope they will show the relevance and potentiality for physical theories of noncommutative lattices. Cambridge, October 1997. Preface to the Online Edition Since this monograph ?rst appeared in October 1997, there has been a host of important developments in the subject. My personal taste would select the following three lines of research but there is much more which I am not mentioning here for lack of space(-time). 1. Applications of noncommutative geometry to string theories.

Details zur Ausgabe

• Verlag: Springer
• Erscheinungsdatum: 2014
• ISBN 10: 3662141094
• ISBN 13: 9783662141090
• Einband: Tapa blanda
• Sprache: Englisch
• Anzahl der Seiten: 224