Path integrals of a particle in a finite interval and on the half-line

TL;DR

This paper develops a method to formulate path integrals for particles confined to finite intervals and half-lines by transforming variables and accounting for boundary reflections through phase factors, linking potential parameters to reflection signs.

Contribution

It introduces a new canonical variable framework enabling path integral formulation on bounded domains, incorporating boundary reflections via phase factors dependent on potential parameters.

Findings

Path integrals can be formulated using transformed variables on finite intervals.

Reflected paths require extending variables to the covering space.

Phase factors for reflections depend on potential parameters.

Abstract

We make use of point transformations to introduce new canonical variables for systems defined on a finite interval and on the half-line so that new position variables should take all real values from $-\infty$ to $\infty$. The completeness of eigenvectors of new momentum operators enables us to formulate time sliced path integrals for such systems. Short time kernels thus obtained require extension of the range of variables to the covering space in order to take all reflected paths into account. Upon this extension we determine phase factors attached to the amplitude for paths reflected at boundaries by taking singularities of the potential into account. It will be shown that the phase factor depends on parameters that characterize the potential; and further that the well-know minus sign in the amplitude for odd times reflection of a particle in a box should be understood as the special case for the corresponding value of the parameter of the potential.