  The structure of the interaction Hamiltonian in the first order $S-$matrix
element of a Dirac particle in an Aharonov-Bohm (AB) field is analyzed and
shown to have interesting algebraic properties. It is demonstrated that as a
consequence of these properties, this interaction Hamiltonian splits both the
incident and outgoing waves in the the first order $S-$matrix into their
$\frac{\Sigma_3}{2}-$components (eigenstates of the third component of the
spin). The matrix element can then be viewed as the sum of two transitions
taking place in these two channels of the spin. At the level of partial waves,
each partial wave of the conserved total angular momentum is split into two
partial waves of the orbital angular momentum in a manner consistent with the
conservation of the total angular momentum quantum number.
