Unitarity in space–time noncommutative field theories

In noncommutative field theories conventional wisdom is that the unitarity is noncompatible with the perturbation analysis when time is involved in the noncommutative coordinates. However, as suggested by Bahns et al. recently, the root of the problem lies in the improper definition of the time-ordered product. In this article, functional formalism of S-matrix is explicitly constructed for the noncommutative φp scalar field theory using the field equation in the Heisenberg picture and proper definition of time-ordering. This S-matrix is manifestly unitary. Using the free spectral (Wightmann) function as the free field propagator, we demonstrate the perturbation obeys the unitarity, and present the exact two particle scattering amplitude for (1+1)-dimensional noncommutative nonlinear Schrödinger model.