Geometric momentum in the Monge parametrization of two dimensional sphere

TL;DR

This paper explores the geometric momentum on a two-dimensional sphere using the Monge parametrization, deriving explicit forms and addressing quantum mechanical commutation relations through Dirac's constrained motion theory.

Contribution

It provides an explicit formulation of geometric momentum on a sphere in Monge coordinates, incorporating Dirac's theory to handle second-class constraints and ensuring invariance and self-adjointness.

Findings

Explicit expression for geometric momentum on a sphere

Inclusion of Dirac's theory for constrained quantum mechanics

Demonstration of invariance and self-adjointness of the momentum

Abstract

A two dimensional surface can be considered as three dimensional shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of two dimensional sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [x_{i},p_{j}]=i \hbar({\delta}_{ij}-x_{i}x_{j}/r^2) rather than [x_{i},p_{j}]=i\hbar{\delta}_{ij} that does not hold true any more. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.