Path integration in the field of a topological defect: the case of dispiration

The motion of a particle in the field of dispiration (due to a wedge disclination and a screw dislocation) is studied by path integration. By formulating a SO(2) x T (1) gauge theory, first we derive the metric, curvature and torsion of the medium of dispiration. Then we carry out explicitly path integration for the propagator of a particle moving in the non-Euclidean medium under the influence of a scalar potential and a vector potential. We also obtain the winding number representation of the propagator by taking the non-trivial topological structure of the medium into account. We extract the energy spectrum and the eigenfunctions from the propagator. Finally we make some remarks for special cases. Particularly, paying attention to the difference between the result of the path integration and the solution of Schrodinger's equation in the case of disclination, we suggest that the Schrodinger equation may have to be modified by a curvature term.