  We study field theories on the noncommutative Minkowski space with
noncommuting time. The focus lies on dispersion relations in quantized
interacting models in the Yang-Feldman formalism. In particular, we compute the
two-point correlation function of the field strength in noncommutative quantum
electrodynamics to second order. At this, we take into account the covariant
coordinates that allow the construction of local gauge invariant quantities
(observables). It turns out that this does not remove the well-known severe
infrared problem, as one might have hoped. Instead, things become worse, since
nonlocal divergences appear. We also show that these cancel in a supersymmetric
version of the theory if the covariant coordinates are adjusted accordingly.
  Furthermore, we study the phi^3 and the Wess-Zumino model and show that the
distortion of the dispersion relations is moderate for parameters typical for
the Higgs field. We also disuss the formulation of gauge theories on
noncommutative spaces and study classical electrodynamics on the noncommutative
Minkowski space using covariant coordinates. In particular, we compute the
change of the speed of light due to nonlinear effects in the presence of a
background field. Finally, we examine the so-called twist approach to quantum
field theory on the noncommutative Minkowski space and point out some
conceptual problems of this approach.
