Impact-Parameter Representations and the Coordinate-Space Description of a Scattering Amplitude

We show how impact-parameter representations with a dependence on final c.m. momentum k, angle $\theta$, and impact parameter $b$ given by $J_0\left(kbsin\theta \right)$ follow naturally when we describe a scattering process in terms of states localized in the transverse plane in coordinate space, or more generally when we analyze it in terms of a Fourier transform with respect to the transverse momentum. We discuss this for the nonrelativistic as well as the relativistic scattering amplitude. Two classes of impact-parameter representations are considered here: one in which the impact parameter is a position coordinate, canonically conjugate to the transverse momentum; and another—a fixed-energy representation—which is obtained by reinterpreting the Fourier transform with respect to the transverse momentum, in which the impact parameter is not a position coordinate, except in the limit of infinite energy. We show that the first kind of impact-parameter representation follows from the description of a scattering process in terms of transient asymptotic states localized in the transverse coordinate space. We point out the distinctive features of each type of representation, and discuss the conditions under which each is valid.