An Introduction to Noncommutative Differential Geometry and its Physical Applications

A significant amount of the differential structure of a smooth manifold can be encoded in the algebra of smooth functions defined on it. A noncommutative geometry is what one obtains when one replaces this algebra by a noncommutative associative algebra. Of particular interest is the case when the algebra is of finite dimension, for example an algebra of matrices. The resulting geometry is rather trivial from the point of view of analysis and can serve as a simple introduction to some of the more elementary aspects of the noncommutative geometries which one can obtain by considering more general infinite-dimensional algebras. It also has certain specific additional properties which makes it well suited to the construction of finite models of space-time. A more or less complete survey of this geometry is given as well as some possible applications to elementary particle physics and field theory. This book arose from the 1994 LMS invited lectures and will be essential to mathematicians and theoretical physicists with an interest in noncommutative geometry.

An introduction to some of the more elementary aspects of non-commutative geometry, with special emphasis on those cases where the structure algebra, which defines the geometry, is an algebra of matrices over the complex numbers. Also included are applications to elementary particle physics. The material is pitched at graduate students, and only some familiarity with ordinary differential geometry and the theory of fibre bundles is assumed. All who wish to know more about this fascinating area of research will find that this book provides an excellent introduction.