Gauge Field, Parity and Uncertainty Relation of Quantum Mechanics on S^1

TL;DR

This paper explores the uncertainty principle, parity, and symmetry breaking for quantum particles on a circle, using a formulation of quantum mechanics on spheres, and discusses challenges in extending these concepts to general manifolds.

Contribution

It applies a recent quantum mechanics formulation on spheres to analyze uncertainty and parity on $S^1$, revealing phenomena like spontaneous symmetry breaking.

Findings

Uncertainty of position on $S^1$ is bounded due to compactness.

Parity symmetry can be spontaneously broken in this setting.

Challenges are identified in formulating quantum mechanics on general manifolds.

Abstract

We consider the uncertainty relation between position and momentum of a particle on $ S^1 $ (a circle). Since $ S^1 $ is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on $ S^D $ (a $D$-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate quantum mechanics on a general manifold.