Noncommutative differential geometry, quantum mechanics and gauge theory

Abstract

We describe a noncommutative differential calculus, introduced in [1], which generalizes the differential calculus of differential forms of E. Cartan. We show that besides the classical (commutative) situation, this differential calculus is well suited to deal with ordinary quantum mechanics. That is quantum mechanics falls in the framework of a noncommutative symplectic geometry. We then introduce the simplest corresponding gauge theories. We show that these theories describe ordinary gauge theories but with multivacua structures which provide a sort of alternative to the Higgs mechanism. Most of this lecture is based on a joint work with R. Kerner and J. Madore [2], [3], [4], [5].