We show that the photon self-energy in quantum electrodynamics on noncommutative R 4 is renormalizable to all orders (both in θ andh) when using the Seiberg-Witten map. This is due to the enormous freedom in the Seiberg-Witten map which represents field redefinitions and generates all those gauge invariant terms in the θ-deformed classical action which are necessary to compensate the divergences coming from loop integrations.

We investigate the quantization of the θ-expanded noncommutative U(1) Yang-Mills action, obtained via the Seiberg-Witten map. As expected we find non-renormalizable terms. The one-loop propagator corrections are gauge independent, and lead us to a unique extention of the noncommutative classical action. We interpret our results as a requirement that also the trace in noncommutative field theory should be deformed.

Abstract. For the noncommutative Yang-Mills field there exist two representations (primitive and covariant) of the (undeformed) group of rigid translations, rotations and dilatations. The SeibergWitten map is the equivalence between both representations on the level of classical noncommutative field theories. In the covariant representation the Yang-Mills field is θ-dependent according to a first-order differential equation and thus can be parametrized by its initial value at θ = 0, i.e. a gauge field living on commutative space-time.

We discuss the different possibilities of constructing the various energy-momentum tensors for noncommutative gauge field models. We use Jackiw's method in order to get symmetric and gauge invariant stress tensors-at least for commutative gauge field theories. The noncommutative counterparts are analyzed with the same methods. The issues for the noncommutative cases are worked out.

We introduce the notion of superoperators on noncommutative R 4 and re-investigate in the framework of superfields the noncommutative Wess-Zumino model as a quantum field theory. In a highly efficient manner we are able to confirm the result that this model is renormalizable to all orders.2 Work supported by The Danish Research Agency. 4 Work supported in part by "Fonds zur Förderung der Wissenschaftlichen Forschung" (FWF) under contract P13125-PHY and P13126-PHY.6 Marie-Curie Fellow.