Impact parameter representations from the point of view of the Poincaré group

Impact parameter representations for the two-particle scattering amplitude are derived by expanding the two-particle state with respect to groups that are isomorphic to the group of motions of the plane $E(2)$. The generators of these groups are expressed in terms of the operators of the total angular momentum and the relative momentum of two particles, As a result, the impact parameter operator and states of two particles with definite impact parameter acquire a meaning in terms of the Poincaré group.