Topological Phase In Gauge Field Theory

Topological phase, due to its particular feature that the efect directly relate to themost fundamental symmetry of physics, is the most important one of a variety of quantumphases. The frst topological phase was predicted by the Israel physicist Y. Aharonov andAmerican physicist D. Bohm in1959, which was called Aharonov-Bohm AB phase efect.Another type of topological phase was proposed by Y. Aharonov and A. Casher in1984,which was called Aharonov-Casher AC phase efect. In this paper, we give out a detaildiscussion on these topological phase efects, as well as the application of dual symmetryin these topological efects.Firstly, we give out a brief review of the gauge symmetry in the classical electromag-netic interaction. Then we discuss the gauge symmetry of electromagnetic interaction inquantum mechanics. Based on these discussions, we review the generation mechanismof topological Aharonov-Bohm efect. Secondly, we discus and reexamine the topologicalAharonov-Casher and Scalar-Aharonov-Bohm (SAB) phase efects which are induced bythe interaction between spin and external electromagnetic felds. The equation of motionof particle with fxed polarization is derived. This equation possesses an invariance whichis similar to the gauge invariance in the electromagnetic interaction. And then this in-variance can induce shift of interference pattern. The generation conditions derived fromthe equation of motion is consistent with experiment. By reexamining the non-relativisticlimit of this equation, we fnd that the conclusion of M. Peshkin and H. J. Lipkin is worrybecause of their incorrect equation of motion. In our interpretation, the topological andnonlocal nature of the AC and SAB phase efects are apparent, and then implies that anew complementarity between locality and non-locality is not necessary when explaining topological phase efect.Finally, we discuss the dual symmetry and dual transformations in classical electro-magnetic theory. Then we discuss the dual relations between the Aharonov-Casher andHe-McKellar-Wilkens topological phases which are related to the electromagnetic interac-tion. Our results show that, when particle has both magnetic and electric dipole moments,the topological phases of these two moments do possess the dual symmetry. We providethe general Lagrangian for the Aharonov-Casher and He-McKellar-Wilkens topologicalphases, and construct a model independent dual relation between the Aharonov-Casherand He-McKellar-Wilkens topological phases by assuming that the electromagnetic inter-action with magnetic monopole violates parity symmetry but consistent with the relativ-ity. The parameter is related to the strength of parity violation in the electromagneticinteraction, and can be used in the study of parity violation and dual symmetry forexperiments.