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

TL;DR

This paper uses path integration to analyze quantum particle motion in a medium with dispiration, revealing topological effects and suggesting modifications to Schrödinger's equation in curved spaces.

Contribution

It provides a detailed path integral formulation for particles in dispiration fields, incorporating topology, curvature, and torsion effects, and discusses potential modifications to quantum equations.

Findings

Derived the metric, curvature, and torsion of dispiration medium.

Explicit path integral for the propagator in a non-Euclidean medium.

Indicated possible need to modify Schrödinger's equation with curvature terms.

Abstract

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 gauging $SO(2) \otimes T(1)$, 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 obtain also 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 Schr\"odinger's equation in the case of disclination, we suggest that Schr\"odinger equation may have to be modified by a curvature term.