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

• URL: http://arxiv.org/abs/2103.03408v2
• Date: Sat, 15 May 2021 13:32:17 GMT
• Title: The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere
• Authors: Z. Li, L. Q. Lai, Y. Zhong, and Q. H. Liu
• Abstract summary: We show that the momentum for the particle on the hypersphere is the geometric one including the gauge potential. We demonstrate that the momentum for the particle on the hypersphere is the geometric one including the gauge potential.
• Score: 0.46664938579243576
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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