Revisiting the Schrödinger–Dirac Equation

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