  Implications of noncommutative field theories with commutator of the
coordinates of the form $[x^{\mu},x^{\nu}]=i \Lambda_{\quad \omega}^{\mu
\nu}x^{\omega}$with nilpotent structure constants are investigated. It is shown
that a free quantum field theory is not affected by noncommutativity, but that
invariance under translations is broken and hence the energy-momentum
conservation is not respected. The new energy-momentum law is expressed by a
Poincar\'e-invariant equation and the resulting kinematics is developed and
applied to the astrophysical puzzle related with the observed violation of the
GZK cutoff.
