Quantum mechanics for three versions of the Dirac equation in a curved   spacetime
  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. 
Mayeul Arminjon and Frank Reifler