Hermiticity and Self-Adjointness in Quantum Mechanics

Abstract

Hamiltonians in the geveralized Feshbach–Villars and Foldy–Wouthuysen representations describing an interaction of a scalar particle with electromagnetic fields in the Minkowski spacetime are self-adjoint and Hermitian (or pseudo-Hermitian) when they are presented in terms of operators of covariant derivatives. When one uses curvilinear coordinates in special relativity, the transition to the canonical momentum operator does not change these properties. When the curvilinear coordinates are applied in general relativity, the corresponding transition to the canonical momentum operator leads to the seeming nonHermiticity of the Hamiltonians. Since the Hamiltonians remain in fact Hermitian, this seeming non-Hermiticity should not be eliminated by any nonunitary transformation.