We discuss the nonabelian world-volume action which governs the dynamics of N coincident Dp-branes. In this theory, the branes’ transverse displacements are described by matrix-valued scalar fields, and so this is a natural physical framework for the appearance of noncommutative geometry. One example is the dielectric effect by which Dp-branes may be polarized into a noncommutative geometry by ...

In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are covered: the one based on derivations and the one based on spectral triples. Ex...

The purpose of these lectures is to survey several aspects of noncommutative geometry, with emphasis on its applicability to particle physics and quantum field theory. By now, it is a commonplace statement that spacetime at short length —or high energy— scales is not the Cartesian continuum of macroscopic experience, and that older methods of working with functions on manifolds must be rethough...

We introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah’s KR-theory and by Tomita’s involution J. It allows us to remove two unpleasant features of the “Connes-Lott” description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincare duality and in the unimodularity condition.

Foreword This book is not just a survey of noncommutative geometry — later NCG; rather, it strives to answer some naive but vital questions: What is the purpose of NCG? What is it good for? Can NCG solve open problems of classical geometry inaccessible otherwise? In other words, why does NCG matter? What is it anyway? Good answer means good examples. A sweetheart of NCG called non-commutative t...

The two existing theories which successfully encode our knowledge of space-time are : • General Relativity • The Standard Model General relativity describes space-time as far as large scales are concerned (cf. Figure 1 and [2] for many more suggestive thoughts and pictures) and is based on the geometric paradigm discovered by Riemann. It replaces the flat (pseudo) metric of Poincaré, Einstein, ...

Our geometric concepts evolved first through the discovery of NonEuclidean geometry. The discovery of quantum mechanics in the form of the noncommuting coordinates on the phase space of atomic systems entails an equally drastic evolution. We describe a basic construction which extends the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and ...

The main result of the paper is Egorov’s theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.

“Extremely” localized wavefunctions in noncommutative geometry have disturbances that are localized to distances smaller than √ θ, where θ is the “area” parameter that measures noncommutativity. In particular, distributions such as the sign function or the Dirac delta function are limiting cases of extremely localized wavefunctions. It is shown that Moyal star products of extremely localized wa...