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Two alternative Dirac equations with gravitation

An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems. One such class is that of the locally-geodesic systems. If a preferred reference frame is available, another class emerges. From the classical Hamiltonian that rules geodesic motion, the correspondence yields two distinct Klein-Gordon (KG) equations and two distinct Dirac equations in a general metric, depending on the class selected. Each of the two Dirac equations can be put in generally-covariant form, transforms the wave function as a four-vector, and differs from the standard (Fock-Weyl) gravitational Dirac equation. One obeys the equivalence principle in the usually-accepted sense, which the Fock-Weyl equation does not.