Motion of a massive fermion field in curved spacetime

This paper presents a comprehensive derivation of the equation of motion for a massive fermion field within a curved spacetime background. The formulation incorporates standard Dirac coupling, along with specific non-minimal gravitational interaction terms characterized by parameters \lambda and \eta. The derivation strictly adheres to the principle of least action, performing variations with respect to the adjoint fermion field. This rigorous approach yields an equation of motion that accounts for both mass and non-minimal couplings to the gravitational field.