The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere

TL;DR

This paper explores the geometric momentum of a particle constrained on a hypersphere, revealing a curvature-induced gauge potential and the associated commutation relations that incorporate both angular momentum and curvature effects.

Contribution

It demonstrates that the geometric momentum includes a gauge potential and obeys specific commutation relations, extending previous understanding of quantum motion on curved hyperspherical surfaces.

Findings

The momentum includes a curvature-induced gauge potential.

The components of geometric momentum obey specific commutation relations.

The generalized angular momentum combines orbital and curvature effects.

Abstract

A particle that is constrained to freely move on a hyperspherical surface in an $N\left( \geq 2\right) $ dimensional flat space experiences a curvature-induced gauge potential, whose form was given long ago (J. Math. Phys. \textbf{34}(1993)2827). We demonstrate that the momentum for the particle on the hypersphere is the geometric one including the gauge potential and its components obey the commutation relations $\left[ p_{i},p_{j}\right] =-i\hbar J_{ij}/r^{2}$, in which $\hbar $ is the Planck's constant, and $p_{i}$ ($i,j=1,2,3,...N$) denotes the $i-$th component of the geometric momentum, and $J_{ij}$ specifies the $ij-$th component of the generalized\textit{\ angular momentum} containing both the orbital part and the coupling of the generators of continuous rotational symmetry group $% SO(N-1)$ and curvature, and $r$ denotes the radius of the $N-1$ dimensional hypersphere.