Discrete time evolution and energy nonconservation in noncommutative physics

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