On relationship between canonical momentum and geometric momentum

TL;DR

This paper explores the relationship between canonical and geometric momentum in higher-dimensional spaces, deriving a hermitian form of the momentum operator along the normal direction using Gaussian normal coordinates.

Contribution

It introduces a decomposition of the gradient operator and formulates a hermitian canonical momentum along the normal direction in terms of mean curvature.

Findings

Derived a hermitian form of the normal canonical momentum

Connected the momentum operator to geometric properties like mean curvature

Showed the surface component of momentum matches geometric momentum

Abstract

Decompositing of $N+1$-dimensional gradient operator in terms of Gaussian normal coordinates $(\xi^{0},\xi^{\mu})$, ($\mu=1,2,3,...,N$) and making the canonical momentum $P_{0}$ along the normal direction $\mathbf{n}$ to be hermitian, we obtain $\mathbf{n}P_{0}=-i\hbar\left( \mathbf{n}\partial _{0}-\mathbf{M}_{0}\right) $ with $\mathbf{M}_{0}$ denoting the mean curvature vector on the surface $\xi^{0}=const.$ The remaining part of the momentum operator lies on the surface, which is identical to the geometric one.