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

TL;DR

This paper investigates how the curvature of a hypersurface causes noncommutativity in the momentum components of a constrained particle, revealing curvature-dependent algebraic relations and geometric effects like rotations and translations.

Contribution

It introduces a local analysis using the generalized Dupin indicatrix to connect hypersurface curvature with momentum noncommutativity and geometric phase effects.

Findings

Noncommutativity relates to products of curvature and angular momentum or momentum.

Noncommutativity induces a rotation operator with an angle anholonomy.

Along normal sectional curves, noncommutativity results in translation and a half-angle rotation.

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.