Dirac Equation in the Presence of a Gravitational Background: Temporal Non Commutative Framework and Vierbein Formalism

In this paper, we study the non commutative quantum dynamics of the spinorial field in the presence of a gravitational background within DeWitt's approach. By employing the vierbein formalism in the temporal state of non commutative four-dimensional space time, we also analyze the first level correction of the solution and Green's function in terms of the Riemann curvature tensor. The effect of the gravitational background is imposed at least up to the first order of the Riemann curvature tensor.