  It has been suggested that one may construct a Lorentz-invariant
noncommutative field theory by extending the coordinate algebra to additional,
fictitious coordinates that transform nontrivially under the Lorentz group.
Integration over these coordinates in the action produces a four-dimensional
effective theory with Lorentz invariance intact. Previous applications of this
approach, in particular to a specific construction of noncommutative QED, have
been studied only in a low-momentum approximation. Here we discuss
Lorentz-invariant field theories in which the relevant physics can be studied
without requiring an expansion in the inverse scale of noncommutativity.
Qualitatively, we find that tree-level scattering cross sections are
dramatically suppressed as the center-of-mass energy exceeds the scale of
noncommutativity, that cross sections that are isotropic in the commutative
limit can develop a pronounced angular dependence, and that nonrelativistic
potentials (for example, the Coloumb potential) become nonsingular at the
origin. We consider a number of processes in noncommutative QED that may be
studied at a future linear collider. We also give an example of scattering via
a four-fermion operator in which the noncommutative modifications of the
interaction can unitarize the tree-level amplitude, without requiring any other
new physics in the ultraviolet.
