Noncommutative Geometry and Quantization

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 rethought in order to handle granular or bubbly spacetime and the matter fields which they support. The question as to what should replace the continuum is an ongoing one. Noncommutative geometry (NCG) offers a general approach to the job of describing the geometrical aspects of nonclassical spaces. The hope is that it can provide a solid framework for studying fundamental interactions and QFT. Here we consider four aspects of this general problem. In Section 1, we discuss the noncommutative tori that have recently emerged as an important model is string theory. Section 2 reviews the general structure of noncommutative geometries based on the spectral triples of Connes. The third section broaches the formulation of quantum field theories over noncommutative spaces, using the example of a 3-torus. In Section 4, we sketch how Hopf algebras provide a link between NCG and renormalization calculations, that illuminates the claim that “QFT is the geometry of the world”.