  Quantum group Fourier transform methods are applied to the study of processes
on noncommutative Minkowski spacetime $[x^i,t]=\imath\lambda x^i$. A natural
wave equation is derived and the associated phenomena of {\it in vacuo}
dispersion are discussed. Assuming the deformation scale $\lambda$ is of the
order of the Planck length one finds that the dispersion effects are large
enough to be tested in experimental investigations of astrophysical phenomena
such as gamma-ray bursts. We also outline a new approach to the construction of
field theories on the noncommutative spacetime, with the noncommutativity
equivalent under Fourier transform to non-Abelianness of the `addition law' for
momentum in Feynman diagrams. We argue that CPT violation effects of the type
testable using the sensitive neutral-kaon system are to be expected in such a
theory.
