HELICITY CONSERVATION IN THE AHARONOV-BOHM SCATTERING OF DIRAC PARTICLES

We show that the helicity operator LAMBDA for a particle in the presen ce of an infinitely thin magnetic flux tube requires, as the Hamiltoni an H, for its complete determination as a self-adjoint operator, the s pecification of boundary conditions (BC's) that have to be chosen out of a four parameter family of admissible ones. To each value of the pa rameters there corresponds a self-adjoint operator with eigenfunctions and eigenvalues determined by the associated BC's. For each choice of the dynamics H we investigate under which conditions the correspondin g BC is also admissible for the helicity LAMBDA. When this happens, an d only when this happens, it is possible for H and LAMBDA to satisfy i dentical BC's. Although their actions formally commute before specific ation of boundary conditions, only identical BC's will ensure effectiv e commutativity, in the sense that they will have a complete set of co mmon eigenfunctions and that LAMBDA will be a conserved quantity. We s how this to be the case only for a special (but large) class of BC's. Our results imply that helicity conservation, although imposing some r estrictions on the choice of the dynamics, does not solve the problem of the indeterminacy in the choice of BC's in the Aharonov-Bohm scatte ring of Dirac particles. Our results also show that it is possible to choose BC's such that both helicity is conserved and the Aharonov-Bohm symmetry (phi --> phi + 1) is preserved, where phi is the magnetic fl ux in natural units.