  Time-space noncommutativity leads to quantisation of time and energy
nonconservation when time is conjugate to a compact spatial direction like a
circle. In this context energy is conserved only modulo some fixed unit. Such a
possibility arises for example in theories with a compact extra dimension with
which time does not commute. The above results suggest striking
phenomenological consequences in extra dimensional theories and elsewhere. In
this paper we develop scattering theory for discrete time translations. It
enables the calculation of transition probabilities for energy nonconserving
processes and has a central role both in formal theory and phenomenology.
  We can also consider space-space noncommutativity where one of the spatial
directions is a circle. That leads to the quantisation of the remaining spatial
direction and conservation of momentum in that direction only modulo some fixed
unit, as a simple adaptation of the results in this paper shows.
