Discrete time evolution and energy nonconservation in noncommutative physics

Time-spacing noncommunitativity leads to quantisation of time [1] and energy non-conservation [2] 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 wit ha 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 had a central role both in formal theory and phenomenology. We can also consider space-space noncommunitativity where one of the spatial directions is a circle. That leads to the quantisation of the remaining spatial direction direction and conservation of momentum in that direction only modulo some fixed unit, as a simple adaptation of the results in this paper shows.