  In flat spacetime, the Dirac equation is the "square root" of the
Klein-Gordon equation in the sense that by applying the square of the Dirac
operator to the Dirac spinor, one recovers the Klein-Gordon equation duplicated
for each component of the spinor. In the presence of gravity, applying the
square of the curved-spacetime Dirac operator to the Dirac spinor does not
yield the curved-spacetime Klein-Gordon equation, but yields, instead, the
Schr\"odinger-Dirac covariant equation. First, we show that the latter equation
gives rise to a generalization to spinors of the covariant Gross-Pitaevskii
equation. Next, we show that while the Schr\"odinger-Dirac equation is not
conformally invariant, there exists a generalization of the equation that is
conformally invariant but which requires a different conformal transformation
of the spinor than the one required by the Dirac equation. The new conformal
factor acquired by the spinor is found to be a matrix-valued factor obeying a
differential equation that involves the Fock-Ivanenko line element. The
Schr\"odinger-Dirac equation coupled to the Maxwell field is then revisited and
generalized to particles with higher electric and magnetic moments while
respecting gauge symmetry. Finally, Lichnerowicz's vanishing theorem in the
conformal frame is also discussed.
