An Introduction to Noncommutative Spaces and their Geometry
Abstract
These lectures notes are an intoduction for physicists to several ideas and applications of noncommutative geometry. The necessary mathematical tools are presented in a way which we feel should be accessible to physicists. We illustrate applications to YangMills, fermionic and gravity models, notably we describe the spectral action recently introduced by Chamseddine and Connes. We also present an introduction to recent work on noncommutative lattices. The latter have been used to construct topologically nontrivial quantum mechanical and field theory models, in particular alternative models of lattice gauge theory. Here is the list of sections: 1. Introduction. 2. Noncommutative Spaces and Algebras of Functions. 3. Noncommutative Lattices. 4. Modules as Bundles. 5. The Spectral Calculus. 6. Noncommutative Differential Forms. 7. Connections on Modules. 8. Field Theories on Modules. 9. Gravity Models. 10. Quantum Mechanical Models on Noncommutative Lattices. Appendices: Basic Notions of Topology. The Gel'fandNaimarkSegal Construction. Hilbert Modules. Strong Morita Equivalence. Partially Ordered Sets. Pseudodifferential Operators
 Publication:

arXiv eprints
 Pub Date:
 January 1997
 arXiv:
 arXiv:hepth/9701078
 Bibcode:
 1997hep.th....1078L
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Quantum Algebra
 EPrint:
 181 + iv pages, Latex, 26 figures included in the source file. A revised and enlarged version has been published in Lecture Notes in Physics: Monographs, m51 (SpringerVerlag, Berlin Heidelberg, 1997) ISBN 3540635092