  The Dirac wave function in a curved spacetime is usually defined as a
quadruplet of scalar fields. It can alternatively be defined as a four-vector
field. We describe these two representations in a common geometrical framework
and we prove theorems that relate together the different representations and
the different choices of connections. In particular, the standard Dirac
equation in a curved spacetime, with any choice of the tetrad field, is
equivalent to a particular realization of the Dirac equation for a vector wave
function, in the same spacetime.
