Regularization and Renormalization of Quantum Field Theories on Noncommutative Spaces

Abstract

We first review regularization methods based on matrix geometry which provide an ultraviolet cut-off for scalar fields respecting the symmetries. Sections of bundles over the sphere can be quantized, too. This procedure even allows to regularize supersymmetry without violating it. Recently, this work was extended to include quantum group covariant regularizations.In a second part recent attempts to renormalize four-dimensional deformed quantum field theory models are reviewed. For scalar models the well-known UV/IR-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, may allow to avoid this problem.There was some hope that circumventing UV/IR-mixing by application of the Seiberg-Witten map one can achieve renormalizability of noncommutative field theories. Although the photon self-energy is renormalizable to all orders, θ-expanded noncommutative QED was shown to be not renormalizable, thus ruling out this approach as well.