Discrete time evolution and energy nonconservation in noncommutative physics

Time-space noncommutativity leads to quantisation of time and energynonconservation when time is conjugate to a compact spatial direction like acircle. In this context energy is conserved only modulo some fixed unit. Such apossibility arises for example in theories with a compact extra dimension withwhich time does not commute. The above results suggest strikingphenomenological consequences in extra dimensional theories and elsewhere. Inthis paper we develop scattering theory for discrete time translations. Itenables the calculation of transition probabilities for energy nonconservingprocesses and has a central role both in formal theory and phenomenology. We can also consider space-space noncommutativity where one of the spatialdirections is a circle. That leads to the quantisation of the remaining spatialdirection and conservation of momentum in that direction only modulo some fixedunit, as a simple adaptation of the results in this paper shows.Comment: 17 pages, LaTex; minor correction