Curvature-induced noncommutativity of two different components of momentum for a particle on a hypersurface

Abstract

As a nonrelativistic particle constrained to remain on an $N-1$ ($N\geq 2$) dimensional hypersurface embedded in an $N$ dimensional Euclidean space, two different components $p_{i}$ and $p_{j}$ ($i,j=1,2,3,...N$) of the Cartesian momentum of the particle are not mutually commutative, and explicitly commutation relations $[p_{i},p_{j}]\left( \neq 0\right) $ depend on products of positions and momenta in uncontrollable ways. The \textit{% generalized} Dupin indicatrix of the hypersurface, a local analysis technique, is utilized to explore the dependence of the noncommutativity on the curvatures on a \textit{local point }of the hypersurface. The first finding is that the noncommutativity can be grouped into two categories; one is the product of a sectional curvature and the angular momentum, and another is the product of a principal curvature and the momentum. The second finding is that, for a small circle lying a \textit{tangential plane} covering the \textit{local point}, the noncommutativity leads to a rotation operator and the amount of the rotation is an angle anholonomy; and along each of the \textit{normal sectional curves} centering the \textit{given point} the noncommutativity leads to a translation plus an additional rotation and the amount of the rotation is one half of the tangential angle change of the arc.