  We present a recent work on the Dirac equation in a curved spacetime. In
addition to the standard equation, two alternative versions are considered,
derived from wave mechanics, and based on the tensor representation of the
Dirac field. The latter considers the Dirac wave function as a spacetime vector
and the set of the Dirac matrices as a third-order tensor. Having the
probability current conserved for any solution of the Dirac equation gives an
equation to be satisfied by the coefficient fields. A positive definite scalar
product is defined and a hermiticity condition for the Dirac Hamiltonian is
derived for a general coordinate system in a general curved spacetime. For the
standard equation, the hermiticity of the Dirac Hamiltonian is not preserved
under all admissible changes of the coefficient fields.
