Revisiting the Schr\"odinger-Dirac equation

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.