Geometric momentum for a particle constrained on a curved hypersurface

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 $N-1$ ($N\ge 2$) 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.

5 pages, no figure

References in corpus (6)

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• Enhanced Quantization: A Primer
• Quantum motion on a torus as a submanifold problem in a generalized Dirac's theory of second-class constraints
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Cited by in corpus (11)

• Geometric effects resulting from square and circular confinements for a particle constrained to a space curve
• Hamiltonian for a particle in a magnetic field on a curved surface in orthogonal curvilinear coordinates
• Geometric Potential Resulting from Dirac Quantization
• The centripetal force law and the equation of motion for a particle on a curved hypersurface
• Generalized Centripetal Force Law and Quantization of Motion Constrained on 2D Surfaces
• The geometric potential of a double-frequency corrugated surface
• "Classical" coherent state generated by curved surface
• General covariant geometric momentum, gauge potential and a Dirac fermion on a two-dimensional sphere
• Sphere on a plane: Two-dimensional scattering from a finite curved region
• Geometric momentum and angular momentum for charge-monopole system
• Generally covariant geometric momentum and geometric potential for a Dirac fermion on a two-dimensional hypersurface