Quantum Mechanics on S^n and Meron Solution

TL;DR

This paper explores how quantum particles on spheres couple to topological gauge fields, revealing that on S^2 the gauge is like a magnetic monopole, while on S^3 and higher spheres it resembles a meron configuration.

Contribution

It demonstrates that the induced gauge potentials on spheres are analogous to monopoles and merons, providing new insights into topological effects in quantum mechanics on curved manifolds.

Findings

Gauge potential on S^2 is like a magnetic monopole.

Gauge potential on S^3 is like a meron.

Generalization to higher spheres S^{2n+1}.

Abstract

A particle in quantum mechanics on manifolds couples to the induced topological gauge field that characterises the possible inequivalent quantizations. For instance, the gauge potential induced on $S^2$ is that of a magnetic monopole located at the center of $S^2$. We find that the gauge potential induced on $S^3$ ($S^{2n+1}$) is that of a meron (generalized meron) also sitting at the center of $S^3$ ($S^{2n+1}$).