arXiv 2103.034082021-03hep-thmath-phmath.MPquant-ph

超球上粒子的曲率诱导规范势和几何动量

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

Z. Li, L. Q. Lai, Y. Zhong, Q. H. Liu

备注：: 7 pages, no figure. Major revision in both scientific content and represetation
发表信息：: Annals of Physics, Volume 432, September 2021, 168566
DOI：: 10.1016/j.aop.2021.168566

摘要：

被约束在$N\left( \geq 2\right) $维平坦空间中的超球面上自由运动的粒子经历曲率诱导规范势，其形式在很久以前就给出了（J. Math.Phys. \textbf{34}（1993）2827）。我们证明了超球上粒子的动量是包含规范势的几何动量，其分量服从对易关系$\left[ p_{i},p_{j}\right] =-i\hbar J_{ij}/r^{2}$，其中$\hbar $是普朗克常数，$p_{i}$是量子力学中的量子力学中的量子（$i,j=1,2,3,...N$）表示几何动量的第$i-$分量，而$J_{ij}$指定了广义\textit{\ angular momentum}的第$ij-$分量，它既包含轨道部分又包含连续旋转对称生成元的耦合组$% SO(N-1)$和曲率，$r$表示$N-1$维超球面的半径。

英文摘要：

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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