Path Integral Formulation of Curved Low Dimensional Space

Abstract

When a low dimensional space has a curvature, there is an effective potential in the Schrödinger equation as a geometrical correction. In this paper, we have shown that a confined space in three dimensional space can be regarded as a curved low (one or two) dimensional space when the thickness of the space multiplied by the Weingarten map of each space is sufficiently smaller than unity. Under the condition, we have also evaluated the effective potential using the path integral method.