Generalized Galilei invariant partial wave expansions of the scattering amplitude for collisions between two particles with arbitrary spin

Abstract

The invariant operators of the Euclidean group E(3) and its chains of subgroups E(3)⊃0(3)⊃O(2) and E (3)⊃E(2)XT_⊥⊃O(2)xT_⊥provide bases of eigenfunctions for the construction of generalized Gal ilei invariant partial wave expansions of the scattering amplitude for non-relativistic collisions between particles of arbitrary spin. These expansions are generalizations of those obtained by Kalnins et.al.¹) for spinless particles. The first chain of groups produces a spherical expansion which is a generalization of the well known helicity formalism. The second chain of groups gives rise to two different cylindrical representations of the scattering amplitude, each one related to one of the two symmetry axes in the collision. The cylindrical expansion associated to the total momentum axis of symmetry is a generalization of the impact parameter eikonal expansion supplemented with an additional expansion in the remaining kinematical variable. Associated to the momentum transfer axis, there is another cylindrical expansion of the scattering amplitude which coincides with the non-relativistic limit of the crossed channel expansion of the relativistic amplitude as shown by Cocho and Mondragôn². In every case, the scattering amplitude and the partial wave amplitude are integral transforms one of the other. The kernels of these transforms are expressed in terms of matrix elements of the group operators appropriate to each case.