  An analysis of the classical-quantum correspondence shows that it needs to
identify a preferred class of coordinate systems, which defines a torsionless
connection. One such class is that of the locally-geodesic systems,
corresponding to the Levi-Civita connection. Another class, thus another
connection, emerges if a preferred reference frame is available. From the
classical Hamiltonian that rules geodesic motion, the correspondence yields two
distinct Klein-Gordon equations and two distinct Dirac-type equations in a
general metric, depending on the connection used. Each of these two equations
is generally-covariant, transforms the wave function as a four-vector, and
differs from the Fock-Weyl gravitational Dirac equation (DFW equation). One
obeys the equivalence principle in an often-accepted sense, whereas the DFW
equation obeys that principle only in an extended sense.
