Quantum mechanics in Riemannian spacetime. I. Generally covariant Schrödinger equation with relativistic corrections

A study is made of the $c^{-2}$ asymptotics ($c$ is the speed of light) of the theory of a complex scalar field in a general Riemannian spacetime; the field interacts with an external electromagnetic field. In a freely falling (Gaussian normal) frame of reference we obtain a generally covariant analog of the Schrödinger equation for a scalar particle in external gravitational and electromagnetic fields with relativistic corrections of arbitrary order. It is shown that allowance for the geometrical variation in time of the phase-space element leads to a Hamiltonian that is (asymptotically) Hermitian with respect to the standard scalar product, and this provides a basis for the Born interpretation of the corresponding wave functions.