Topological Phases and Their Duality in Electromagnetic and Gravitational Fields

The duality found by Aharonov and Casher for topological phases in the electromagnetic field is generalized to an arbitrary linear interaction. This provides a heuristic principle for obtaining a new solution of the field equations from a known solution. This is applied to the general relativistic Sagnac phase shift due to the gravitational field in the interference of mass or energy around a line source that has angular momentum and the dual phase shift in the interference of a spin around a line mass. These topological phases are treated both in the linearized limit of general relativity and the exact solutions for which the gravitational sources are cosmic strings containing torsion and curvature, which do not have a Newtonian limit. As is well known, some of Yakir Aharonov's most famous contributions concern topological phases due to the electromagnetic field. It is therefore fitting on this occasion of his sixtieth birthday for me to present him with some observations concerning these phases, which generalize naturally to the gravitational field. In particular, I shall examine the duality between the Aharonov-Bohm (AB) phase [1] and the phase shift in the interference of a magnetic moment in an electric field [2] which was found by Aharonov and Casher (AC) [3]. I shall show, by means of the linearized limit and an exact solution of the gravitational field equations, that both these phases have gravitational analogs and they satisfy this duality. In section 1, I shall briefly summarize the phase shifts in the interference of a charge and a magnetic dipole (at low energies) due to the electromagnetic field. These phase shifts reveal, respectively, U (1) and SU (2) gauge field aspects of the electromagnetic field. But these two aspects are not independent: The SU (2) connection which gives the dipole phase shift depends on the electric and magnetic fields and as such are derived from the electromagnetic connection that gives the U (1) AB phase shift. It is nevertheless amusing to see a charged particle with a magnetic moment, such as an electron, interacting with an electromagnetic field as if it is a U (1) × SU (2) gauge field. Two topological phase shifts due to electric and manetic fields corresponding to two U (1) subgroups of SU (2) will be reviewed. The