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

Abstract

For a particle that is constrained to freely move on a hypersurface, the curvature of the surface can induce a gauge potential; and for a particle on the hypersphere, the gauge potential derived from the generalized angular momentum algebra on it has been known long before (J. Math. Phys. 34(1993Phys. 34( )2827. We demonstrate that the momentum for the particle on the hyperspherecan be the geometric one which obey commutation relations [pi, pj ] = −i Jij /r 2 , in which is the Planck's constant, and pi (i = 1, 2, 3, ...N ) symbolizes the i−th component of the geometric momentum, and Jij specifies the ij−th component of the angular momentum containing the spin-curvature coupling, and r denotes the radius of the N − 1 dimensional hypersphere.