Quantum Field Theory on Noncommutative Spaces

The old dream to obtain a consistent description unifying quantum mechanics and relativity is still far from being realized. The problems connected to Quantum Field Theory are one manifestation of that problem. Although the unification of electro-, magnetic-, weak and strong interactions is a big step in the direction of a consistent treatment, it is generally believed that gravity has to be taken into account. There is therefore no doubt that our concept of spacetime breaks down at tiny small distances, and differential manifolds should be quantized. Such modifications are the starting point of our project. In recent years, there has been considerable effort to study field theoretical models on various noncommutative spaces. Both scalar and gauge field theories have been studied. Our research will focus on four different types of noncommutative spaces: the canonically quantized spacetime, fuzzy spheres, q-deformed fuzzy spheres and Kappa-deformed spaces. The latter spaces have the advantage that they allow for a generalized (deformed) notion of symmetry, whereas canonical spacetime does not. In general quantum effects for these field theories are not yet under control. In particular, a phenomenon known as "UV-IR mixing" threatens perturbative renormalizability. The consistent implementation of generalized symmetries is also not well understood. The aim of the proposed project is two-fold: First, we want to obtain a better understanding of the quantization of field theories on quantized spaces, using both renormalization group techniques as well as the particular simplifications which occur on fuzzy spaces. On the fuzzy sphere, gauge theory may be defined in a completely nonperturbative way in terms of matrix models which are invariant under U(N). One of the goals is to take advantage of these methods in order to show that a well- defined classical limit exists, and that these models are renormalizable. Second, we plan to study the implementation of generalized symmetries which exist on Kappa-deformed and q-deformed spaces in field theory. Lagrangian models for gauge theories which are symmetric under a Kappa-deformed respectively q-deformed symmetry are not yet fully formulated and understood. For the kappa-deformed space, this can be accomplished using the Seiberg-Witten map. The formulation of gauge theory on q-deformed fuzzy spheres in terms of matrix models may also provide new insights.