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An Introduction to Noncommutative Spaces and Their Geometries

  • Book
  • © 1997

Overview

Authors:
  1. Giovanni Landi

    1. Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, Trieste, Italy

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  • This book is a comprehensible introduction for physicists to A. Connes'approach to noncommutative geometry.

Part of the book series: Lecture Notes in Physics Monographs (LNPMGR, volume 51)

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Table of contents (11 chapters)

  1. Front Matter

    Pages I-XV
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  2. Introduction

    Pages 1-5

  3. Noncommutative Spaces and Algebras of Functions

    Pages 7-20

  4. Projective Systems of Noncommutative Lattices

    Pages 21-58

  5. Modules as Bundles

    Pages 59-68

  6. A Few Elements of K-Theory

    Pages 69-81

  7. The Spectral Calculus

    Pages 83-103

  8. Noncommutative Differential Forms

    Pages 105-121

  9. Connections on Modules

    Pages 123-132

  10. Field Theories on Modules

    Pages 133-150

  11. Gravity Models

    Pages 151-163

  12. Quantum Mechanical Models on Noncommutative Lattices

    Pages 165-169

  13. Back Matter

    Pages 171-207
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Keywords

About this book

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.

Authors and Affiliations

  • Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, Trieste, Italy

    Giovanni Landi

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