Eikonal evolution and gluon radiation

We give a simple quantum mechanical formulation of the eikonal propagation approximation, which has been heavily used in recent years in problems involving hadronic interactions at high energy. This provides a unified framework for several approaches existing in the literature. We illustrate this scheme by calculating the total, elastic, inelastic and diffractive DIS cross sections, as well as gluon production in high energy hadronic collisions. From the $q\overline{q}g$ component of the DIS cross sections, we straightforwardly derive low x evolution equations for inelastic and diffractive DIS distribution functions. In all calculations, we provide all order $1/N$ corrections to the results existing in the literature.