Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

Abstract

We first review regularization methods based on matrix geometry and show how an ultraviolet cut-off for scalar fields respecting symmetries results. Sections of bundles over the sphere can be quantized too.This procedure even allows to regularize supersymmetry without violating it.This work was extended recently to include quantum group covariant regularizations.

In a second part recent attempts to renormalize fourdimensional deformed quantum field theory models is reviewed. For scalar models the well-known IR-UV mixing does not allow to use standard techniques.The same applies to the Yang-Mills model in four dimensions.Only additional symmetry, as it occurs in the Wess-Zumino model, allows to avoid this problem.

Nevertheless there is some hope that the Yang-Mills model can be handled too. We used the Seiberg-Witten map to transform the noncommutative gauge field to a commutative one and used the degree of freedom of this map to obtain counter terms for the renormalization procedure. Finally a derivation of the Seiberg-Witten map from natural requirements is skechted.