Observable Dirac-Type Singularities in Berry’s Phase and the Monopole

A three-dimensional generalization of the sign-change ( $\pi$ phase shift) rule for adiabatic cycles of spin-1/2 or two-state wave functions encircling a degeneracy in the parameter space of the Hamiltonian yields a Dirac-type singularity wherein any closed circuit of the adiabatic cycle in which the degeneracy is “looped” results in an observable $\pm 2\pi$ phase shift. It is concluded that an interferometer loop similarly taken around a magnetic monopole of strength $n/2\mathrm{}$ yields an observable $\pm 2n\pi$ phase shift, $n$ being an integer.