Geometric momentum for a particle constrained on a curved hypersurface

TL;DR

This paper introduces a new quantization scheme for particles constrained on curved hypersurfaces, deriving a momentum dependent on mean curvature and resolving the geometric potential for spheres, revealing a new SO(N,1) symmetry.

Contribution

It proposes a strengthened canonical quantization method with new commutation relations and derives the geometric momentum and potential for curved surfaces, including a novel symmetry for spheres.

Findings

Derived momentum depends on mean curvature.

Resolved the geometric potential for spherical surfaces.

Identified a new SO(N,1) symmetry in spherical motion.

Abstract

A strengthened canonical quantization scheme for the constrained motion on a curved hypersurface is proposed with introduction of the second category of fundamental commutation relations between Hamiltonian and positions/momenta, whereas those between positions and moments are categorized into the first. As an $N-1$ ($N\geq2$) dimensional hypersurface is embedded in an N dimensional Euclidean space, we obtain the proper momentum that depends on the mean curvature. For the surface is the spherical one, a long-standing problem on the form of the geometric potential is resolved in a lucid and unambiguous manner, which turns out to be identical to that given by the so-called confining potential technique. In addition, a new dynamical group SO(N,1) symmetry for the motion on the sphere is demonstrated.