On relationship between canonical momentum and geometric momentum

Abstract

Decompositing of N + 1-dimensional gradient operator in terms of Gaussian normal coordinates (ξ 0 , ξ µ ), (µ = 1, 2, 3, ..., N ) and making the canonical momentum P 0 along the normal direction n to be hermitian, we obtain nP 0 = −i (n∂ 0 − M 0 ) with M 0 denoting the mean curvature vector on the surface ξ 0 = const. The remaining part of the momentum operator lies on the surface, which is identical to the geometric one.