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

Abstract

A particle that is constrained to freely move on a hyperspherical surface in an $N\left(\ge 2\right)$ 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 $\left[p_i,p_j\right]=-i\hslash J_{ij}/r^2$, in which $\hslash$ 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 and curvature, and $r$ denotes the radius of the $N-1$ dimensional hypersphere.

7 pages, no figure. Major revision in both scientific content and represetation

References in corpus (2)

• Non-Abelian gauge field theory of the spin-orbit interaction and a perfect spin filter
• The centripetal force law and the equation of motion for a particle on a curved hypersurface

Cited by in corpus (1)

• Generally covariant geometric momentum and geometric potential for a Dirac fermion on a two-dimensional hypersurface