Quantum mechanics for three versions of the Dirac equation in a curved spacetime

1 Quantum Mechanics in a gravitational field ◮ Quantum effects in the classical gravitational field are observed, e.g. in neutron interferometry [1, 2]. All of the effects observed until now had been previously predicted by using the non-relativistic Schrödinger equation in the Newtonian gravity potential [2, 3, 4], and are still described by this same non-relativistic approximation. ◮ However, gravity is currently described by relativistic theories with curved spacetime. ◮ The usual way to write the wave equations of quantum mechanics (QM) in a curved spacetime is by covariantization (this is connected with the equivalence principle): at any given event X, the sought-for equation in a curved spacetime should coincide with the flat-spacetime version in coordinates where the connection vanishes at X and g µν (X) = η µν .