On relationship between canonical momentum and 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 $\mathbf{n}{P}_0=-i\hslash \left(\mathbf{n}{\partial }_0-{\mathbf{M}}_0\right)$ 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.

7 pages