General-covariant quantum mechanics in Riemannian space-time III. The Dirac particle

Abstract

A general covariant analog of standard nonrelativistic quantum mechanics with relativistic corrections is constructed for the Dirac particle in a normal geodesic frame in general Riemannian space-time. Not only the Pauli equation with a Hermitian Hamiltonian and the pre-Hilbert structure of the space of its solutions, but also matrix elements of the Hermitian operators of momentum, (curvilinear) spatial coordinates, and spin of the particle, are deduced, as a general-covariant asymptotic approximation in c⁻² (c is the velocity of light), to their naturally determined general-relativistic pre-images. It is shown that the Pauli equation Hamiltonian, generated by the Dirac equation, is unitary-equivalent to the energy operator generated by the metric energymomentum tensor of the spinor field. Commutation and other properties of the observables associated with variation in the geometrical background of quantum mechanics are briefly discussed.