  Starting from an interpretation of the classical-quantum correspondence, we
derive the Dirac equation by factorizing the algebraic relation satisfied by
the classical Hamiltonian, before applying the correspondence. This derivation
applies in the same form to a free particle, to one in an electromagnetic
field, and to one subjected to geodesic motion in a static metric, and leads to
the same, usual form of the Dirac equation--in special coordinates. To use the
equation in the static-gravitational case, we need to rewrite it in more
general coordinates. This can be done only if the usual, spinor transformation
of the wave function is replaced by the 4-vector transformation. We show that
the latter also makes the flat-space-time Dirac equation Lorentz-covariant,
although the Dirac matrices are not invariant. Because the equation itself is
left unchanged in the flat case, the 4-vector transformation does not alter the
main physical consequences of that equation in that case. However, the equation
derived in the static-gravitational case is not equivalent to the standard
(Fock-Weyl) gravitational extension of the Dirac equation.
